text:
\section{Introduction}

In 1992, Kurc \cite{Kurc} presented a relation between the best dominated approximation and order continuity and monotonicity properties. In the paper \cite{CieKolPan}, authors answered the question about the full characterization of the local monotonicity structure and order continuity of Banach lattices and applications in the best dominated approximation. The next motivating research was published in \cite{CieKolPlu}, where there has been established among others a connection between the best dominated approximation and the Kadec-Klee property for global convergence in measure in Banach function spaces. Recently, in view of the previous investigation there appeared many results \cite{Cies-geom,CKKP,CiesKamPluc,CieKolPluSKM,HKL} devoted to exploration of the global and local monotonicity and rotundity structure of Banach spaces applicable in the best approximation problems. The main inspiration for this article appeared in paper \cite{Cies-JAT}, where there has been introduced a new type of the best dominated approximation with respect to the Hardy-Littlewood-P\'olya relation $\prec$. It is worth mentioning that author has investigated an application of strict  $K$-monotonicity and $K$-order continuity to the best dominated approximation problems in the sense of the Hardy-Littlewood-P\'olya relation. In the spirit of the previous articles, the essential purpose of this paper is to focus on the complete criteria for $K$-order continuity in symmetric spaces.

It is necessary to recall the significant paper \cite{ChDSS}, in which there has been shown a correlation between strict $K$-monotonicity and global convergence in measure of a sequence of the decreasing rearrangements in symmetric spaces. The next intention of this paper is to find a local version of a correspondence between strict $K$-monotonicity and global convergence in measure of a sequence of the maximal functions in symmetric spaces.

The present article is organized as follows. Preliminaries contain all needed terminologies, which will be used later. Here we also recall an earlier result, which play a crucial role in our further exploration.

Section 3 is dedicated to strict $K$-monotonicity in a symmetric space $E$. We start our investigation with the auxiliary result proving an existence of at most two upper bounds of $\mathcal{M}(x,\tau,\epsilon)$ the family of the decreasing rearrangements bounded by an element $x\in{E}$ with respect to the Hardy-Littlewood-P\'olya relation $\prec$. Then, in view of the above result, we present an application of a point of lower $K$-monotonicity to global convergence in measure of a sequence from the cone of all nonnegative decreasing elements in $E$.

In the last section 4 we answer the key question whether an existence of a point of $K$-order continuity in a symmetric space $E$ on $[0,\infty)$ guarantees that $E$ is not embedded in $L^1[0,\infty)$. Next, we discuss the full criteria for a point of $K$-order continuity in a symmetric space $E$. Namely, we show a complete correlation between a point of $K$-order continuity and a point of order continuity in a symmetric space $E$ under an additional assumption of the fundamental function $\phi$ in case when $I=[0,\infty)$. Finally, we describe a relationship between a point of order continuity and an $H_g$ point in a symmetric space $E$.

\section{Preliminaries}

Assume that $\mathbb{R}$, $\mathbb{R}^+$ and $\mathbb{N}$ are the sets of reals, nonnegative reals and positive integers, respectively.  For a Banach space $(X,\norm{\cdot}{X}{})$ we denote by $S(X)$ (resp.
$B(X))$ the unit sphere (resp. the closed unit ball). A nonnegative function $\phi$ defined on $\mathbb{R}^+$ is said to be \textit{quasiconcave} if $\phi(t)$ is increasing and $\phi(t)/t$ is decreasing on $\mathbb{R}^+$ and also $\phi(t)=0\Leftrightarrow{t=0}$. Let us denote as usual by $\mu$ the Lebesgue measure on $I=[0,\alpha)$, where $\alpha =1$ or $\alpha =\infty$, and by $L^{0}$ the set of all (equivalence classes of) extended real valued Lebesgue measurable functions on $I$. We employ the notation $A^{c}=I\backslash A$ for any measurable set $A$. Given a Banach lattice $(E,\Vert \cdot \Vert _{E})$ is said to be a \textit{Banach
function space} (or a \textit{K\"othe space}) if it is a sublattice of $L^{0}
$ satisfying the following conditions
\begin{itemize}
\item[(1)] If $x\in L^0$, $y\in E$ and $|x|\leq|y|$ a.e., then $x\in E$ and $%
\|x\|_E\leq\|y\|_E$.
\item[(2)] There exists a strictly positive $x\in E$.
\end{itemize}
Additionally, to simplify the notation in the whole paper it is used the symbol $E^{+}={\{x \in E:x \ge 0\}}$.

An element $x\in E$ is called a \textit{point of order continuity} if for any
sequence $(x_{n})\subset{}E^+$ such that $x_{n}\leq \left\vert x\right\vert
$ and $x_{n}\rightarrow 0$ a.e. we have $\left\Vert x_{n}\right\Vert
_{E}\rightarrow 0.$ A Banach function space $E$ is said to be \textit{order continuous
}(shortly $E\in \left( OC\right) $) if any element $x\in{}E$ is a point of order continuity (see \cite{LinTza}). We say that a Banach function space $E$ has the \textit{Fatou property} if for any $\left( x_{n}\right)\subset{}E^+$, $\sup_{n\in \mathbb{N}}\Vert x_{n}\Vert
_{E}<\infty$ and $x_{n}\uparrow x\in L^{0}$, then we get $x\in E$ and $\Vert x_{n}\Vert _{E}\uparrow\Vert x\Vert
_{E}$. Unless it is said otherwise, in the whole article it is considered that $E$ has the Fatou property.

An element $x\in{E}$ is called an $H_g$ \textit{point} in $E$ if for any sequence $(x_n)\subset{E}$ such that $x_n\rightarrow{x}$ globally in measure and $\norm{x_n}{E}{}\rightarrow\norm{x}{E}{}$, then $\left\Vert x_n-x\right\Vert _{E}\rightarrow{0}$. Let us recall that the space $E$ has the \textit{Kadec-Klee property for global convergence in measure} if each element $x\in{E}$ is an $H_g$ point in $E$.

The \textit{distribution function} for any function $x\in L^{0}$ is given by
\begin{equation*}
d_{x}(\lambda) =\mu\left\{ s\in [ 0,\alpha) :\left\vert x\left(s\right) \right\vert >\lambda \right\},\qquad\lambda \geq 0.
\end{equation*}
For each element $x\in L^{0}$ we define its \textit{decreasing rearrangement} by
\begin{equation*}
x^{\ast }\left( t\right) =\inf \left\{ \lambda >0:d_{x}\left( \lambda
\right) \leq t\right\}, \text{ \ \ } t\geq 0.
\end{equation*}
 We use the convention $x^{*}(\infty)=\lim_{t\rightarrow\infty}x^{*}(t)$ if $\alpha=\infty$ and $x^*(\infty)=0$ if $\alpha=1$. For any function $x\in L^{0}$ we define the \textit{maximal function} of $x^{\ast }$ by
\begin{equation*}
x^{\ast \ast }(t)=\frac{1}{t}\int_{0}^{t}x^{\ast }(s)ds.
\end{equation*}
Given $x\in L^{0}$, it is commonly known that $x^{\ast }\leq x^{\ast \ast },$ $x^{\ast \ast }$ is decreasing, continuous and subadditive. For more details of $d_{x}$, $x^{\ast }$ and $x^{\ast \ast }$ the reader is referred to see \cite{BS, KPS}.

Let us recall that two functions $x,y\in{L^0}$ are called \textit{equimeasurable} (shortly $x\sim y$) if $d_x=d_y$. A Banach function space $(E,\Vert \cdot \Vert_{E}) $ is said to be \textit{symmetric} or \textit{rearrangement invariant} (r.i. for short) if whenever $x\in L^{0}$ and $y\in E$ such that $x \sim y,$ then $x\in E$ and $\Vert x\Vert_{E}=\Vert y\Vert _{E}$. For a symmetric space $E$ we define $\phi_{E}$ its \textit{fundamental function} given by $\phi_{E}(t)=\Vert\chi_{(0,t)}\Vert_{E}$ for any $t\in [0,\alpha)$ (see \cite{BS}). Given $x,y\in{}L^{1}+L^{\infty }$ we define the \textit{Hardy-Littlewood-P\'olya relation} $\prec$ by
\begin{equation*}
x\prec y\Leftrightarrow x^{\ast \ast }(t)\leq y^{\ast \ast }(t)\text{ for
all }t>0.\text{ }
\end{equation*}

A symmetric space $E$ is said to be $K$-\textit{monotone} (shortly $E\in(KM)$)
if for any $x\in L^{1}+L^{\infty}$ and $y\in E$ with $x\prec y,$ then $x\in E$ and $\Vert x\Vert_{E}\leq \Vert y\Vert _{E}.$ It is well known that a symmetric space is $K$-monotone if and only if $E$ is exact
interpolation space between $L^{1}$ and $L^{\infty }.$ Let us also recall that a symmetric space $E$ equipped with an order continuous norm or with the Fatou property is $K$-monotone (see \cite{KPS}).

Given $x\in{E}$ is said to be a \textit{point of lower $K$-monotonicity} of $E$ (shortly an $LKM$ \textit{point}) if for any $y\in{E}$, $x^*\neq{y^*}$ with $y\prec{x}$, then $\norm{y}{E}{}<\norm{x}{E}{}$. Let us mention that given a symmetric space $E$ is called \textit{strictly $K$-monotone} (shortly $E\in(SKM)$) if any element of $E$ is an $LKM$ point.

An element $x\in{E}$ is called a \textit{point of $K$-order continuity} of $E$ if for any sequence $(x_n)\subset{E}$ with $x_n\prec{x}$ and $x_n^*\rightarrow{0}$ a.e. we have $\norm{x_n}{E}{}\rightarrow{0}$. A symmetric space $E$ is said to be $K$-\textit{order continuous} (shortly $E\in\left(KOC\right)$) if any element $x$ of $E$ is a point of $K$-order continuity. We refer the reader for more information to see \cite{ChDSS,Cies-JAT,Cies-geom,CieKolPluSKM}.

For any quasiconcave function $\phi$ on $I$ the Marcinkiewicz function space $M_{\phi}^{(*)}$ (resp. $M_{\phi}$) is a subspace of $L^0$ such that
$$\norm{x}{M_{\phi}^{(*)}}{}=\sup_{t>0}\{x^*(t)\phi(t)\}<\infty$$
$${\left(\textnormal{resp. }\norm{x}{M_{\phi}}{}=\sup_{t>0}\{x^{**}(t)\phi(t)\}<\infty\right).}$$
Clearly, $M_{\phi}\overset{1}{\hookrightarrow}M_{\phi}^{(*)}$. It is well known that the Marcinkiewicz space $M_{\phi}^{(*)}$ (resp. $M_{\phi}$) is a r.i. quasi-Banach function space (r.i. Banach function space) with the fundamental function $\phi$ on $I$. It is worth reminding that for any symmetric space $E$ with the fundamental function $\phi$ we have $E\overset{1}{\hookrightarrow}M_{\phi}$ (for more details see \cite{BS,KPS}).

Let us recall auxiliary result proved in \cite{Cies-JAT} which will be useful in our further investigation.
\begin{lemma}\label{lem:x**=0}
	Let $x\in{L^1+L^\infty}$ and $x^*(\infty)=0$, then $x^{**}(\infty)=0$.
\end{lemma}

\section{local approach to strict $K$-monotonicity in symmetric spaces}

Now, we introduce the special set which is similar to a notion that was presented in \cite{KPS}. Let $E$ be a symmetric space and $x\in{E}$, $x=x^*$. Define for any $\epsilon>0$ and $\tau>0$,
\begin{equation*}
\mathcal{M}(x,\tau,\epsilon)=\left\{y\in{E}:y=y^*,y\prec{x},\int_{0}^{\tau}y(s)ds+\epsilon\leq\int_{0}^{\tau}x(s)ds\right\}.
\end{equation*}
The next result shows that for any collection $\mathcal{M}(x,\tau,\epsilon)$ there exist at most two elements that are upper bounds of a given family less than $x$ with respect to the relation $\prec$. The proof of the below result is quite long and required completely different techniques than the proof stated in Corollary 2.2 \cite{ChDSS}. Although the case $1$ in the proof follows from the case $2$, we present the whole details for the sake of the reader's convenience and because of the construction that might arouse the reader's interest.

\begin{proposition}\label{prop:2:upper:bound}
Let $x\in{L^0}$, $x=x^*$, $x^*(\infty)=0$ and $\tau>0$, $\epsilon>0$. Then, there exist $\epsilon_1>0$, $\tau_1\in(0,\tau)$, $z\in\mathcal{M}(x,\tau-\tau_1,\epsilon_1)$ and $w\in\mathcal{M}(x,\tau+\tau_1,\epsilon_1)$ such that for any $y\in\mathcal{M}(x,\tau,\epsilon)$ we have $y\prec{z}$ or $y\prec{w}$.
\end{proposition}

\begin{proof}
 Let $y\in\mathcal{M}(x,\tau,\epsilon)$. Define for any decreasing function $u\in{L^0}$  and $t>0$,
\begin{equation*}
\phi_u(t)=\int_{0}^{t}u(s)ds.
\end{equation*}
Since $x^*(\infty)=0$, by monotonicity of functions $x$ and $y$ it is clear that $\phi_x$ and $\phi_y$ are concave, increasing and not affine on $I$. By assumption we have $\phi_y\leq\phi_x$ on $I$ and also $\phi_y(\tau)\leq\phi_x(\tau)-\epsilon$. Let $\gamma$ be an intersection of $\phi_x$ and $p(t)=\phi_x(\tau)-\epsilon$ and let $\beta$ be an intersection of $\phi_x$ and $q(t)=t(\phi_x(\tau)-\epsilon)/\tau$. Denote
\begin{equation*}
\xi=\frac{\phi_x(\beta)-\phi_x(\gamma)}{\beta-\gamma}=\frac{1}{\beta-\gamma}\int_{\gamma}^{\beta}x^*\quad\textnormal{and}\quad{\phi(t)=\xi(t-\gamma)+\phi_x(\gamma)}.
\end{equation*}
Now, we show the proof in cases.\\
\textit{Case $1.$} Assume that there exists $t\in(\gamma,\beta)$ such that $\phi(t)<\phi_x(t)$. Then, by concavity of $\phi_x$ for any $t\in(\gamma,\beta)$ we have $\phi(t)<\phi_x(t)$. Define
\begin{equation*}
z=x^*\chi_{[0,\gamma)\cup[\beta,\infty)}+\xi\chi_{[\gamma,\beta)}.
\end{equation*}
Since $\xi$ is an average value of $x^*$ on $(\gamma,\beta)$, by monotonicity of $x^*$ we observe $z=z^*$, $z\prec{x}$ and $z\neq{x}$. Moreover, by concavity of $\phi_y$ we notice that a tangent line of $\phi_y$ at $\tau$ has a slope $\eta\in[0,(\phi(\tau)-\epsilon)/\tau]$ and for any $t\in{I}$,
\begin{equation*}
\phi_y(t)\leq{\eta(t-\tau)+\phi_x(\tau)-\epsilon}.
\end{equation*}
Hence, for any $t\in[\gamma,\beta]$ we have
\begin{equation*}
\phi_y(t)\leq{\eta(t-\tau)+\phi_x(\tau)-\epsilon}\leq{\xi(t-\tau)+\phi_x(\gamma)}=\phi(t)=\phi_z(t).
\end{equation*}
Consequently, since $\phi_x=\phi_z$ on $[0,\gamma]\cup{[\beta,\infty)}$, by assumption $y\prec{x}$ we get $y\prec{z}$.\\
\textit{Case $2$.} Assume that for any $t\in[\gamma,\beta]$, $\phi_x(t)=\phi(t)$. Denote
\begin{equation*}
\gamma_0=\inf\{t>0:\phi_x(t)=\phi(t)\}.
\end{equation*}
Clearly, $\gamma_0\leq{\gamma}$. We claim that $\gamma_0>0$. Indeed, if we suppose that $\gamma_0=0$, then $\phi_x=\phi$ on $[0,\beta]$ and so $\phi_x(0)=-\xi\gamma+\phi_x(\gamma)=0$. Hence, by definition of $\xi$ and by assumption that $\beta$ is the intersection of $\phi_x$ and $q$ we have $$\xi=\frac{\phi_x(\gamma)}{\gamma}=\frac{\phi_x(\beta)}{\beta}=\frac{\phi_x(\tau)-\epsilon}{\tau}.$$
On the other hand, since $\phi_x$ is affine on $[\gamma_0,\beta]$ it follows that $\xi=\phi_x(\tau)/\tau$. Therefore, we obtain a contradiction which proves our claim. Now, without loss of generality we may suppose that $\epsilon>0$ is small enough such that there exist $\gamma_1>0$ and $\beta_1>0$ the intersections of the functions $\phi_x$ and $\phi-\epsilon$. Observe $0<\gamma_1<\gamma_0<\gamma<\beta<\beta_1$. Consider temporary the slope $\eta$ of the tangent line of $\phi_y$ at $\tau$ is $\eta\in[\xi,(\phi_x(\tau)-\epsilon)/\tau]$. Define
\begin{equation*}
z=x^*\chi_{[0,\gamma_1)\cup[\tau,\infty)}+\frac{\phi_x(\tau)-\phi_x(\gamma_1)}{\tau-\gamma_1}\chi_{[\gamma_1,\tau)}.
\end{equation*}
Then, since $\frac{\phi_x(\tau)-\phi_x(\gamma_1)}{\tau-\gamma_1}$ is the average value of $x^*$ on $(\gamma_1,\tau)$, by definition of $\gamma_0$ and by assumption $\phi_x$ is affine on the interval $(\gamma_0,\beta)$ and by the concavity of $\phi_x$ it follows that $z=z^*$, $z\prec{x}$ and $z\neq{x}$. Furthermore, we have for every $t\in[\gamma_1,\tau]$,
\begin{equation*}
\phi_y(t)\leq{\eta(t-\tau)+\phi_x(\tau)-\epsilon}\leq\phi_z(t).
\end{equation*}
Thus, since $\phi_x=\phi_z$ on $[0,\gamma_1]\cup[\tau,\infty)$, by assumption $y\prec{x}$ we conclude $y\prec{z}$, whenever $\eta\in[\xi,(\phi_x(\tau)-\epsilon)/\tau]$. Now, assume that $\eta\in[0,\xi)$. Define
\begin{equation*}
w=x^*\chi_{[0,\gamma)\cup[\beta_1,\infty)}+\frac{\phi_x(\beta_1)-\phi_x(\gamma)}{\beta_1-\gamma}\chi_{[\gamma,\beta_1)}.
\end{equation*}
Since $\frac{\phi_x(\beta_1)-\phi_x(\gamma)}{\beta_1-\gamma}$ is the average value of $x^*$ on $[\gamma,\beta_1]$, by definition of $\beta_1$ and by concavity of $\phi_x$ we get $w=w^*$, $w\prec{x}$ and $w\neq{x}$. Notice that for any $t\in[\gamma,\beta_1]$,
\begin{equation*}
\phi_y(t)\leq{\eta(t-\tau)+\phi_x(\tau)-\epsilon}\leq\phi_w(t).
\end{equation*}
Moreover, $\phi_x=\phi_w$ on $[0,\gamma]\cup[\beta_1,\infty)$ and by assumption $y\prec{x}$ we obtain $y\prec{w}$ for any $\eta\in[0,\xi)$ which completes case $2$. \\
Finally, combining both cases, by construction of $z$ and $w$ it is easy to see that there exist $\tau_1\in(0,\tau)$ and $\epsilon_1>0$ such that $z\in\mathcal{M}(x,\tau-\tau_1,\epsilon_1)$ and $w\in\mathcal{M}(x,\tau+\tau_1,\epsilon_1)$.
\end{proof}

\begin{theorem}\label{thm:conv:in:meas}
Let $E$ be a symmetric space. Assume that $x\in{E}$ is an $LKM$ point, $x^*(\infty)=0$ and $(x_n)\subset{E}$. If $x_n\prec{x}$ and $\norm{x_n}{E}{}\rightarrow\norm{x}{E}{}$, then $x_n^{**}\rightarrow{x}^{**}$ and $x_n^*\rightarrow{x^*}$ globally in measure.
\end{theorem}

\begin{proof}
Let $\phi$ be the fundamental function of $E$. Since $x_n\prec{x}$ for any $n\in\mathbb{N}$, by symmetry and the Fatou property of $E$ and by Theorem 4.6 in \cite{BS} it follows that
\begin{equation*}
x_n^*(t)\phi(t)\leq\norm{x_n}{E}{}\leq\norm{x}{E}{}
\end{equation*}
for any $t\in{I}$ and $n\in\mathbb{N}$. Hence, by Helly's Selection Theorem in \cite{STV} passing to subsequence and relabelling if necessary there is $z\in{E}$ such that $z=z^*$ and $x_n^*$ converges to $z^*$ a.e. on $I$. It is easy to notice that for all $0<t<s<\alpha$,
\begin{equation}\label{equ:1:cim}
\int_{t}^{s}x_n^*\rightarrow\int_{t}^{s}z^*.
\end{equation}
We claim that for any $t>0$,
\begin{equation}\label{equ:2:cim}
\int_0^{t}x_n^*\rightarrow\int_0^{t}x^*.
\end{equation}
Indeed, if it is not true then there exist $\tau>0$ and $\epsilon>0$ and a sequence $(n_k)\subset\mathbb{N}$ such that for all $k\in\mathbb{N}$,
\begin{equation*}
\int_0^{\tau}x_{n_k}^*\leq\int_0^{\tau}x^*-\epsilon.
\end{equation*}
Consequently, by Proposition \ref{prop:2:upper:bound} there are $\epsilon_1>0$, $\tau_1>0$ and $y\in\mathcal{M}(x,\tau-\tau_1,\epsilon_1)$ and $w\in\mathcal{M}(x,\tau+\tau_1,\epsilon_1)$ such that for any $k\in\mathbb{N}$,
\begin{equation*}
x_{n_k}\prec{y}\quad\textnormal{or}\quad{x_{n_k}\prec{w}}.
\end{equation*}
Since $y^*\neq{x}^*$, $w^*\neq{x^*}$ and $y\prec{x}$, $w\prec{x}$, by assumption that $x$ is an $LKM$ point we get for every $k\in\mathbb{N}$, $$\norm{x_{n_k}}{E}{}\leq\max\{\norm{y}{E}{},\norm{w}{E}{}\}<\norm{x}{E}{}.$$
Thus, by assumption $\norm{x_{n_k}}{E}{}\rightarrow\norm{x}{E}{}$ we obtain a contradiction which provides our claim \eqref{equ:2:cim}. Now, combining conditions \eqref{equ:1:cim} and \eqref{equ:2:cim} we have $x^*=z^*$ a.e. on $I$. Therefore, $x_n^*$ converges to $x^*$ a.e. on $I$. Since $x^*(\infty)=0$, it is easy to prove that $x_n^*$ converges to $x^*$ globally in measure. Indeed, there exists $t_0>0$ such that $x^*(t)<\epsilon/2$ for any $t\geq{t_0}$. Without loss of generality we may assume that $$|x_n^*(t_0)-x^*(t_0)|<\epsilon/2$$
for large enough $n\in\mathbb{N}$. In consequence, by monotonicity of a decreasing rearrangement for all $t\geq{t_0}$ and large enough $n\in\mathbb{N}$ we obtain
\begin{equation*}
x_n^*(t)\leq{x_n^*(t_0)}\leq\abs{x_n^*(t_0)-x^*(t_0)}{}{}+x^*(t_0)<\epsilon.
\end{equation*}
Hence, for any $t\geq{t_0}$ and for large enough $n\in\mathbb{N}$ we get
\begin{equation*}
\abs{x_n^*(t)-x^*(t)}{}{}<\epsilon
\end{equation*}
and since $x_n^*$ converges to $x^*$ locally in measure we conclude global convergence in measure. Moreover, by Lemma \ref{lem:x**=0} we have $x^{**}(\infty)=0$. Hence, there is $t_0>0$ such that for all $t\geq{t_0}$ we have $x^{**}(t)<{\epsilon}$. Since $x_n\prec{x}$ for any $n\in\mathbb{N}$ and by condition \eqref{equ:2:cim} it follows that $x_n^{**}$ converges to $x^{**}$ globally in measure.
\end{proof}

\section{$K$-order continuity in symmetric spaces}

In this section we study a full criteria for $K$-order continuity in symmetric spaces. We start our investigation with an equivalent condition for $K$-order continuity in a symmetric space $E$. This notion was discussed in paper \cite{DSS} for a separable symmetric space with the Fatou property. We study $K$-order continuity in more general case and present a simple example of the order continuous rearrangement invariant space with the Fatou property that is not $K$-order continuous, showing that Proposition 4.2 in \cite{DSS} fails in general case.

\begin{lemma}
Let $E$ be a symmetric space and $x_n,x\in{E}$ for all $n\in\mathbb{N}$ and let:
\begin{itemize}
	\item[$(i)$] If $x_n\prec{x}$ and $x_n^*\rightarrow{0}$ a.e., then $\norm{x_n}{E}{}\rightarrow{0}.$
	\item[$(ii)$] If $x_n\prec{x}$ and $x_n\rightarrow{0}$ globally in measure, then $\norm{x_n}{E}{}\rightarrow{0}.$
\end{itemize}
Then, $(i)\Rightarrow(ii)$. If $x^*(\infty)=0$ then $(ii)\Rightarrow(i)$.
\end{lemma}

\begin{proof}
$(i)\Rightarrow(ii)$. Let assumption be satisfied. Then, by property 2.11 in \cite{KPS} it follows that $x_n^*\rightarrow{0}$ a.e. Consequently, since $x_n\prec{x}$ for any $n\in\mathbb{N}$, by symmetry of $E$ and by condition $(i)$ we get $\norm{x_n}{E}{}\rightarrow{0}$.\\
$(ii)\Rightarrow(i)$. Let $x_n\prec{x}$ for all $n\in\mathbb{N}$ and $x_n^*\rightarrow{0}$ a.e. and let $x^*(\infty)=0$. Then, by Lemma 3.1 in \cite{Cies-JAT} we get $x^{**}(\infty)=0$. Hence, since
\begin{equation*}
x_n^*(t)\leq{x_n^{**}(t)}\leq{x^{**}(t)}
\end{equation*}
for any $t>0$ and $n\in\mathbb{N}$, we easily observe that $x_n^*\rightarrow{0}$ globally in measure. Therefore, since $x_n^*\prec{x}$ for any $n\in\mathbb{N}$, by symmetry of $E$ and by condition $(ii)$ we complete the proof.
\end{proof}

Immediately, by definition of $K$-order continuity and by the above lemma as well as by Lemma 2.5 in \cite{CieKolPan} in case when $E$ is order continuous we conclude the following corollary.

\begin{corollary}
	Let $E$ be a symmetric space and $x_n,x\in{E}$ for all $n\in\mathbb{N}$. If for any $x\in{E}$ we have $x^*(\infty)=0$ or $E$ is order continuous then the following condition are equivalent.
	\begin{itemize}
	\item[$(i)$] $E$ is $K$-order continuous.
    \item[$(ii)$] If $x_n\prec{x}$ and $x_n\rightarrow{0}$ globally in measure, then $\norm{x_n}{E}{}\rightarrow{0}.$
	\end{itemize}
\end{corollary}

Now we answer the question about a correspondence between a point of $K$-order continuity in a symmetric space $E$ and an embedding $E\hookrightarrow{L^1[0,\infty)}$.

\begin{lemma}\label{lem:2:KOC}
Let $E$ be a symmetric space on $I=[0,\infty)$ and let $\phi$ be the fundamental function of $E$. If $x\in{E}\setminus\{0\}$ is a point of $K$-order continuity, then $\lim_{t\rightarrow\infty}{\phi(t)}/{t}=0.$ Additionally, if $x^*(\infty)=0$ we have $\lim_{t\rightarrow\infty}\phi(t)x^{**}(t)=0.$
\end{lemma}

\begin{proof}
Let $t_x>0$ be such that $x^*(t_x)>0$. Define $$y_n=\frac{x^*(t_x)}{n}\chi_{[0,nt_x)}$$ for any $n\in\mathbb{N}$. Clearly,  $y_n=y_n^*\prec{}y_1^*\leq{x^*}$ and $\int_{0}^{\infty}y_n^*=x^*(t_x)t_x<\infty$ for any $n\in\mathbb{N}$. Thus, since $y_n^*\rightarrow{0}$ a.e. and by assumption that $x$ is a point of $K$-order continuity we obtain
\begin{equation*}
\norm{y_n}{E}{}={x^*(t_x)}\frac{\phi(nt_x)}{n}={x^*(t_x)t_x}\frac{\phi(nt_x)}{nt_x}\rightarrow{0}\quad\textnormal{as}\quad{n\rightarrow\infty}.
\end{equation*}
Hence, $\phi(s)/s\rightarrow{0}$ as $s\rightarrow\infty$.
Next, assume that $x^*(\infty)=0$. Denote for any $n\in\mathbb{N}$,
\begin{equation*}
x_n=x^{**}(n)\chi_{[0,n)}.
\end{equation*}
Then, we have $x_n=x_n^*\prec{x}$ for all $n\in\mathbb{N}$. Since $x^*(\infty)=0$, by Lemma \ref{lem:x**=0} it follows that $x_n^*\rightarrow{0}$ a.e. on $I$. Finally, according to assumption that $x$ is a point of $K$-order continuity we get
$\norm{x_n}{E}{}=x^{**}(n)\phi(n)\rightarrow{0}$ as $n\rightarrow\infty$.
\end{proof}

\begin{theorem}
Let $E$ be a symmetric space on $I=[0,\infty)$. If a point $x\in{E}$ is a point of $K$-order continuity, then $E$ is not embedded in $L^1[0,\infty)$.
\end{theorem}

\begin{proof}
Let $\phi$ be a fundamental function of a symmetric space $E$ on $I=[0,\infty)$. Then, we have $\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}>0$ if and only if $\phi(t)\approx{t}$ for large enough $t>0$. Let us mention that the fundamental function $\phi$ of a symmetric space $E$ on $I=[0,\infty)$ satisfies $\phi(t)\approx{t}$ for large enough $t>0$ if and only if $E\overset{C}{\hookrightarrow}{L^1[0,\infty)}$. Hence, by assumption that $x$ is a point of $K$-order continuity, applying Lemma \ref{lem:2:KOC} we finish the proof.
\end{proof}

In the next results we investigate a connection between order continuity and $K$-order continuity in symmetric spaces.

\begin{lemma}\label{lem:KOC=>OC}
	Let $E$ be a symmetric space. If $x\in{E}$ is a point of $K$-order continuity and $x^*(\infty)=0$, then $x$ is a point of order continuity.
\end{lemma}

\begin{proof}
	Let $(x_n)\subset{E}^+$ with $x_n\rightarrow{0}$ a.e. and $x_n\leq\abs{x}{}{}$. Immediately, by the property of the maximal function we obtain $x_n\prec{x}$. Since $x^*(\infty)=0$ and $x_n\leq\abs{x}{}{}$ for all $n\in\mathbb{N}$, by property $2.12$ in \cite{KPS} it follows that $x_n^*\rightarrow{0}$ a.e. Hence, by assumption that $x$ is a point of $K$-order continuity, we have $\norm{x_n}{E}{}\rightarrow{0}$.
\end{proof}

We show that the reverse conclusion of Lemma \ref{lem:KOC=>OC} in general is not true. Namely, we prove that any element in $L^1[0,\infty)$ is not point of $K$-order continuity. Moreover, the following remark yields that Proposition 4.2 in \cite{DSS} is not true in case when $I=[0,\infty)$.

\begin{remark}
	It is well known that each element $x$ of $L^1[0,\infty)$ is a point of order continuity (see \cite{LinTza}). Consider $x=\chi_{[0,1)}$ and $x_n=\frac{1}{n}\chi_{[0,n)}$ for any $n\in\mathbb{N}$. It is easy to see that $x_n\rightarrow{0}$ and $x_n\prec{x}$ for every $n\in\mathbb{N}$. On the other hand, we can easily show $\norm{x_n}{L^1}{}=1$ for any $n\in\mathbb{N}$, which concludes that $x$ is no point of $K$-order continuity. Hence, it is easy to observe that any $y\in{L^1}\setminus\{0\}$ is no point of $K$-order continuity. Indeed, finding $\beta, \gamma\in\mathbb{R}^+$ such that $\beta\chi_{(0,\gamma)}\prec{y^*}$ and proceeding analogously as above with $x$ we finish our investigation.
\end{remark}

In the next example, we show that the assumption $x^*(\infty)=0$ in Lemma \ref{lem:KOC=>OC} is necessary and cannot be omitted.

\begin{example}
We consider the Marcinkiewicz function space $M_{\phi}^{(*)}$ with $\phi(t)=1-\frac{1}{1+t}$ on $I=[0,\infty)$. Define $x=\chi_{[0,\infty)}$. We claim that $x$ is a point of $K$-order continuity and it is no point of order continuity in $M_{\phi}^{(*)}$. It is easy to see that $x^*(\infty)=1$ and $\norm{x}{M_{\phi}^{(*)}}{}=1$. By Lemma 2.5 \cite{CieKolPan} it follows that $x$ is no point of order continuity in $M_{\phi}^{(*)}$. Let $(x_n)\subset{M_{\phi}^{(*)}}$ be such that $x_n\prec{x}$ and $x_n^*\rightarrow{0}$ a.e. In view of the fact that $\norm{\cdot}{M_{\phi}^{(*)}}{}$ satisfies the triangle inequality with a constant $C>0$, we observe for all $n\in\mathbb{N}$,
\begin{align}\label{equ:1:rem:2}
\norm{x_n}{M_{\phi}^{(*)}}{}&\leq{}C\norm{x_n^*\chi_{[0,1]}}{M_{\phi}^{(*)}}{}+C\norm{x_n^*\chi_{(1,\infty)}}{M_{\phi}^{(*)}}{}\\
&\leq{C}\norm{x_n^*\chi_{[0,1]}}{M_{\phi}^{(*)}}{}+{C}x_n^*(1)\norm{x}{M_{\phi}^{(*)}}{}\nonumber.
\end{align}
By assumption $x_n\prec{x}$ and by definition of $x$ and right-continuity of $x^*$ we get $x_n^*(t)\leq{1}$ for all $n\in\mathbb{N}$ and $t\in[0,\infty)$. Thus, by Bolzano-Weierstrass theorem (see \cite{Royd}), passing to subsequence and relabelling if necessary we may assume that for each $n\in\mathbb{N}$ there exists $t_n\in[0,1]$ such that $\lim_{n\rightarrow\infty}t_n=t_0$ and
\begin{equation}\label{equ:2:rem:2}
\norm{x_n^*\chi_{[0,1]}}{M_{\phi}^{(*)}}{}=\sup_{0<t<1}\left\{x_n^*(t)\phi(t)\right\}\leq{}x_n^*(t_n)\left(1-\frac{1}{1+t_n}\right)+\frac{1}{n}
\end{equation}
for all $n\in\mathbb{N}$. Since $x_n^*$ is decreasing for any $n\in\mathbb{N}$, if $t_0>0$, then it is easy to notice that for large enough $n\in\mathbb{N}$,
\begin{equation*}
x_n^*(t_n)\left(1-\frac{1}{1+t_n}\right)\leq{}x_n^*\left(\frac{t_0}{2}\right)\left(1-\frac{1}{1+t_n}\right)\leq{}x_n^*\left(\frac{t_0}{2}\right),
\end{equation*}
whence, by assumption $x_n^*\rightarrow{0}$ a.e. and by conditions \eqref{equ:1:rem:2} and \eqref{equ:2:rem:2}, this yields that $\norm{x_n}{M_{\phi}^{(*)}}{}\rightarrow{0}$. Now, suppose $t_0=0$. Then, $t_n\rightarrow{0}$ and since $x_n^*\leq{1}$ for all $n\in\mathbb{N}$ we have
\begin{equation*}
x_n^*(t_n)\left(1-\frac{1}{1+t_n}\right)\leq{}\left(1-\frac{1}{1+t_n}\right)\rightarrow{0}.
\end{equation*}
Hence, proceeding analogously as in previous case we complete the proof.
\end{example}

In the next results we investigate a complete characterization of $K$-order continuity in symmetric spaces.

\begin{theorem}
	Let $E$ be a symmetric space on $I=[0,\alpha)$, where $\alpha=1$ or $\alpha=\infty$, with the fundamental function $\phi$ and let $x\in{E}$. Then, the following conditions are equivalent.
	\begin{itemize}
		\item[$(i)$]  $x$ is a point of order continuity and in case when $\alpha=\infty$, $$\phi(s)x^{**}(s)\rightarrow{0}\quad\textnormal{as}\quad{}s\rightarrow\infty.$$
		\item[$(ii)$] $x$ is a point of $K$-order continuity and in case when $\alpha =\infty$, $$x^*(\infty)=0.$$
	\end{itemize}
\end{theorem}

\begin{proof}
$(ii)\Rightarrow(i)$. It follows immediately from Lemmas \ref{lem:2:KOC} and \ref{lem:KOC=>OC}.\\
$(i)\Rightarrow(ii)$. We prove only the theorem in case when $\alpha=\infty$, because the proof in the case when $\alpha=1$ is similar.
Let $(x_n)\subset{E}$ and $x_n^*\rightarrow{0}$ a.e., $x_n\prec{x}$. Clearly, we may assume $x\neq{0}$. First, we claim that $x_n^*$ converges to zero globally in measure. Indeed, since for any $t>0$ and $n\in\mathbb{N}$,
\begin{equation}\label{equ:0:LLUKM=>KOC}
x_n^*(t)\leq{x_n^{**}(t)}\leq{x^{**}(t)},
\end{equation}
by Lemma 2.5 in \cite{CieKolPan} and Lemma \ref{lem:x**=0} for any $\epsilon>0$ there is $t_\epsilon>0$ such that for all $t\geq{t_\epsilon}$ and $n\in\mathbb{N}$ we have
\begin{equation}\label{equ:1:LLUKM=>KOC}
\max\{x_n^*(t),x^*(t)\}\leq\frac{\epsilon}{2}.
\end{equation}
Hence, since $x_n^*\rightarrow{0}$ locally in measure we get
\begin{align*}
\mu\left(t:x_n^*(t)\geq\epsilon\right)=\mu\left(t\in[0,t_\epsilon]:x_n^*(t)\geq\epsilon\right)\rightarrow{0},
\end{align*}
which proves our claim. Since $x^*\neq{0}$, there exists $t_0\in(0,t_\epsilon)$ such that $x^*(t_0)>0$. Define $\delta_0=x^*(t_0)/2$ and for any $n\in\mathbb{N}$,
\begin{equation*}
M_n=\left\{t\in[0,t_0]:x^*(t)\leq{x_n^*(t)}\right\}.
\end{equation*}
Then, by monotonicity of $x^*$ it is easy to see that for any $t\in[0,t_0]$,
\begin{equation*}
x^*(t)\geq{}x^*(t_0)>\delta_0.
\end{equation*}
Therefore, for any $n\in\mathbb{N}$ we obtain
\begin{equation*}
M_n\subset\left\{t:x_n^*(t)>\delta_0\right\},
\end{equation*}
whence, by the claim $x_n^*\rightarrow{0}$ globally in measure we get $\mu(M_n)\rightarrow{0}$. Moreover, by assumption that $x_n^*\rightarrow{0}$ a.e., we may suppose without loss of generality that $x_n^*(t_0)\rightarrow{0}.$ Clearly, for any $n\in\mathbb{N}$ we observe
\begin{equation*}
(x_n^*-x^*)^+\chi_{[t_0,t_\epsilon]}\leq{x_n^*}\chi_{[t_0,t_\epsilon]}\leq{x_n^*(t_0)}\chi_{[t_0,t_\epsilon]}.
\end{equation*}
Hence, by symmetry of $E$ it is easy to notice that
\begin{equation*}
\norm{(x_n^*-x^*)^+\chi_{[t_0,t_\epsilon]}}{}{}\leq{x_n^*(t_0)}\norm{\chi_{[t_0,t_\epsilon]}}{}{}
\end{equation*}
for each $n\in\mathbb{N}$. Consequently, since ${x_n^*(t_0)}$ converges to zero, this implies
\begin{equation}\label{equ::part:1:conv}
\norm{(x_n^*-x^*)^+\chi_{[t_0,t_\epsilon]}}{}{}\rightarrow{0}.
\end{equation}
Now, according to condition \eqref{equ:0:LLUKM=>KOC} we have
\begin{equation*}
(x_n^*-x^*)^+\leq{x_n^*}\prec{x^*}\quad\textnormal{and}\quad(x^*-x_n^*)^+\leq{x^*}
\end{equation*}
for every $n\in\mathbb{N}$. In consequence, by Hardy's lemma \cite{BS} it follows that
\begin{equation*}
\int_{0}^{t}((x_n^*-x^*)^+)^*\chi_{[0,\mu(M_n)]}\leq\int_{0}^{t}x^*\chi_{[0,\mu(M_n)]}
\end{equation*}
for all $n\in\mathbb{N}$. Thus, by Proposition 1.1 \cite{ChSu} for any $t>0$ and $n\in\mathbb{N}$ we conclude
\begin{align*}
\left((x_n^*-x^*)^+\chi_{M_n}\right)^{**}(t)&=\frac{1}{t}\int_{0}^{t}\left((x_n^*-x^*)^+\chi_{M_n}\right)^*\\
&\leq\frac{1}{t}\int_{0}^{t}((x_n^*-x^*)^+)^*\chi_{[0,\mu(M_n)]}\\
&\leq\left(x^*\chi_{[0,\mu(M_n)]}\right)^{**}(t)\leq{x}^{**}(t).
\end{align*}
Hence, since $E$ is a symmetric space and
$$(x_n^*-x^*)^+\chi_{M_n}=(x_n^*-x^*)^+\chi_{[0,t_0]}$$
for a.a. on $I$ and for any $n\in\mathbb{N}$, we have
\begin{equation*}
\norm{(x_n^*-x^*)^+\chi_{[0,t_0]}}{E}{}=\norm{(x_n^*-x^*)^+\chi_{M_n}}{E}{}\leq\norm{x^*\chi_{[0,\mu(M_n)]}}{E}{}
\end{equation*}
for each $n\in\mathbb{N}$. In consequence, since $x^*\chi_{[0,\mu(M_n)]}\leq{x^*}$ and $\mu(M_n)\rightarrow{0}$, by assumption that $x$ is a point of order continuity and by Lemma 2.6 \cite{CieKolPan} it follows
\begin{equation*}
\norm{(x_n^*-x^*)^+\chi_{[0,t_0]}}{E}{}\rightarrow{0},
\end{equation*}
whence, by condition \eqref{equ::part:1:conv} and by the triangle inequality of the norm in $E$ we obtain
\begin{equation}\label{equ:6:LLUKM=>KOC}
\norm{(x_n^*-x^*)^+\chi_{[0,t_\epsilon]}}{E}{}\rightarrow{0}.
\end{equation}
Furthermore, since $\min\{x_n^*,x^*\}\leq{x^*}$ and $\min\{x_n^*,x^*\}\rightarrow{0}$ a.e., using analogous argument as previously we conclude
\begin{equation*}
\norm{\min\{x_n^*,x^*\}}{E}{}\rightarrow{0}.
\end{equation*}
Therefore, since $x_n^*=\min\{x_n^*,x^*\}+(x_n^*-x^*)^+$ for any $n\in\mathbb{N}$, by condition \eqref{equ:6:LLUKM=>KOC} we get
\begin{equation}\label{equ:8:LLUKM=>KOC}
\norm{x_n^*\chi_{[0,t_\epsilon]}}{E}{}\rightarrow{0}.
\end{equation}
Now, we need to only show
\begin{equation}\label{equ:second:part:conv}
\norm{x^*_n\chi_{[t_\epsilon,\infty)}}{E}{}\rightarrow{0}.
\end{equation}
Define for any $n\in\mathbb{N}$,
\begin{equation*}
y_n=\left(\frac{1}{n}\int_{0}^{n}x^*\right)\chi_{[0,n)}+x^*\chi_{[n,\infty)}.
\end{equation*}
It is obvious that $y_n=y_n^*$ for all $n\in\mathbb{N}$. We claim that $y_n\prec{x}$ for any $n\in\mathbb{N}$. In fact, by monotonicity of the maximal function of $x^*$, for any $n\in\mathbb{N}$ and $t\leq{n}$ we have
\begin{equation*}
y_n^{**}(t)=\frac{1}{t}\int_{0}^{t}\left(\frac{1}{n}\int_{0}^{n}x^*\right)\chi_{[0,n)}=\frac{1}{n}\int_{0}^{n}x^*\leq{x}^{**}(t).
\end{equation*}
Next, taking $n\in\mathbb{N}$ and $t>n$ by definition of $y_n$ we get immediately $y_n^{**}(t)=x^{**}(t)$ which proves our claim. Since $x^*(\infty)=0$, by Lemma \ref{lem:x**=0} it follows that $y_n^*\rightarrow{0}$ a.e. on $I$. Moreover, for each $n\in\mathbb{N}$ we have
\begin{equation*}
\norm{y_n}{E}{}\leq{\phi(n)}x^{**}(n)+\norm{x^*\chi_{[n,\infty)}}{E}{}.
\end{equation*}
In consequence, since $x$ is a point of order continuity, by Lemma 2.6 in \cite{CieKolPan} and by assumption that $\phi(n)x^{**}(n)\rightarrow{0}$ as $n\rightarrow\infty$ it follows that
\begin{equation}\label{equ:spec:seq:conv}
\norm{y_n}{E}{}\rightarrow{0}.
\end{equation}
Since $x_n^*\rightarrow{0}$ a.e., we may assume without loss of generality that $x_n^*(t_\epsilon)\rightarrow{0}$. Moreover, we may find a subsequence $(n_k)\subset\mathbb{N}$ such that for any $k\in\mathbb{N}$,
\begin{equation*}
x_{n_k}^*(t_\epsilon)\leq\frac{1}{k}\int_{0}^{k}x^*.
\end{equation*}
Fix $k\in\mathbb{N}$. Then, for any $t\in[0,k)$ we get
\begin{equation*}
\left(x_{n_k}^*\chi_{[t_\epsilon,\infty)}\right)^*(t)\leq{}x_{n_k}^*(t_\epsilon)\leq{y_k}^*(t).
\end{equation*}
Furthermore, since $y_k^{**}(t)=x^{**}(t)$ for all $t\geq{}k$ and by assumption $x_n\prec{x}$ for any $n\in\mathbb{N}$, considering $t\geq{}k$ we observe
\begin{equation*}
\left(x_{n_k}^*\chi_{[t_\epsilon,\infty)}\right)^{**}(t)\leq{}x_{n_k}^{**}(t)\leq{x^{**}(t)}={y_k}^{**}(t).
\end{equation*}
Hence, we obtain $x_{n_k}^*\chi_{[t_\epsilon,\infty)}\prec{y_k}$ for each $k\in\mathbb{N}$. In consequence, by symmetry of $E$ this yields that
\begin{equation*}
\norm{x_{n_k}^*\chi_{[t_\epsilon,\infty)}}{E}{}\leq\norm{y_k}{E}{}
\end{equation*}
for any $k\in\mathbb{N}$. Thus, by condition \eqref{equ:spec:seq:conv} and by Lemma 2.3 in \cite{ChDSS} we prove condition \eqref{equ:second:part:conv}. Finally, according to conditions \eqref{equ:8:LLUKM=>KOC} and \eqref{equ:second:part:conv} we get the end of the proof.
\end{proof}

The above theorem implies immediately the following result.

\begin{corollary}\label{coro:KOC}
		Let $E$ be a symmetric space on $I=[0,\alpha)$, where $\alpha=1$ or $\alpha=\infty$, with the fundamental function $\phi$. Then, the following conditions are equivalent.
	\begin{itemize}
		\item[$(i)$]  $E$ is order continuous and in case when $\alpha=\infty$, for any $x\in{E}$, $$\phi(s)x^{**}(s)\rightarrow{0}\quad\textnormal{as}\quad{}s\rightarrow\infty.$$
		\item[$(ii)$] $E$ is $K$-order continuous and in case when $\alpha =\infty$, for any $x\in{E}$, $$x^*(\infty)=0.$$
	\end{itemize}
\end{corollary}

Now, we present an example of a nontrivial symmetric space that is $K$-order continuous.

\begin{example}
	 Assuming that $\phi$ is a quasiconcave function on $I$ with $\inf_{t>0}{\phi(t)}=0$ and $\sup_{t>0}{\frac{t}{\phi(t)}}=\infty$, by Theorem 1.3 in \cite{KamHJLee} it is well known that the subspace of all points of order continuity in the Marcinkiewicz space $M_\phi$ coincides with
	\begin{equation*}
	(M_\phi)_a=\left\{y\in{M_\phi}:\lim\limits_{t\rightarrow{0^+,\infty}}\phi(t)y^{**}(t)=0\right\}\neq\{0\}.
	\end{equation*}
	Finally, according to Corollary \ref{coro:KOC} we observe that $(M_\phi)_a$ is $K$-order continuous.
\end{example}

In the next result we establish an essential correspondence between an $H_g$ point and a point of order continuity in symmetric spaces under an additional assumption.

\begin{proposition}\label{pro:Hg=>OC}
Let $E$ be a symmetric space and let $x\in{E}$, $x^*(\infty)=0$. If $x$ is an $H_g$ point, then $x$ is a point of order continuity in $E$.
\end{proposition}

\begin{proof}
Let $(x_n)\subset{E^+}$, $x_n\leq\abs{x}{}{}$ and $x_n\rightarrow{0}$ a.e. Since $x^*(\infty)=0$, by property 2.12 in \cite{KPS} it follows that $x_n^*(t)\rightarrow{0}$ for all $t>0$. Moreover, by Proposition 1.7 \cite{BS} we have $x_n^*\leq{x^*}$ a.e. and consequently it is easy to show that $x_n^*\rightarrow{0}$ in measure. Hence, we get
$0\leq{x^*-x_n^*\leq{x^*}}$ a.e. for any $n\in\mathbb{N}$ and ${x^*-x_n^*\rightarrow{x^*}}$ in measure. Therefore, it is obvious that $\norm{x^*-x_n^*}{E}{}\leq\norm{x^*}{E}{}$ for any $n\in\mathbb{N}$. Furthermore, by assumption that $x$ is an $H_g$ point, in view of Theorem 3.3 \cite{CieKolPlu} it follows that $x^*$ is an $H_g$ point. Then, by Theorem 3.8 \cite{CieKolPlu} and by the Fatou property we conclude
\begin{equation*}
\norm{x^*-x_n^*}{E}{}\rightarrow\norm{x^*}{E}{}.
\end{equation*}
Thus, since $x^*$ is an $H_g$ point, by symmetry of $E$ we get $\norm{x_n}{E}{}\rightarrow{0}$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Slip into something a little more shimmery with the new Kyra Mule from Katy Perry! The conspicuously stylish Kyra Mule features a wrinkled metallic textile upper with soft faux leather lining, lightly-padded footbed for comfort, and rubber outsole with miniature wrapped heel for a slight lift. Available online at Journeys.com!


For the third consecutive year, Nice Things That Happen in East Meadow is sponsoring a school supplies drive to benefit the Interfaith Nutritional Network in Hempstead.
Josh Sobel, a sixth-grade student at Woodland Middle School, was eager to collect school supplies for children in need.
What began as a Facebook page devoted to highlighting the positive aspects of East Meadow became a conduit for charitable causes and community outreach. Now, Nice Things That Happen in East Meadow is sponsoring its third school supplies drive to benefit the Mary Brennan Interfaith Nutritional Network in Hempstead.
The drive began when Ben Diamond, of East Meadow, reached out to Todd Weinstein, the founder of the Facebook page, and asked if he could use the page as a way to help out the community. At the time, the Mary Brennan INN was seeking help garnering school supplies for those in need.
Each year, the INN provides a number of students and their parents with a hot meal and other necessities, said Dana Lopez, its the director of marketing and communications.
She added, "Through the incredible generosity of community groups like Nice Things that Happen in East Meadow, we are able to properly prepare those students with essential school supplies that will ease the anxiety of not having the correct resources and ultimately arm them for success."
The most needed school supplies items include backpacks, #2 pencils, pens, highlighters, markers, 24-pack crayons, pencil cases, pencil sharpeners, erasers, scissors, glue sticks, marble notebooks, pocket folders, calculators, student planners and loose leaf paper. You could also order your items through the INN's online drive at www.yougivegoods.com/theinn.
This year, those who contribute will receive a token for free ice cream at Stew Leonard's in East Meadow.
The drive goes through Labor Day, but the sooner items are in, the sooner they could be provided to those who need them. For more information, contact Todd Weinstein, pictured right, at (516) 637-3258.


The datasets and their corresponding ground-truth used in tranScriptorium can be downloaded from this web page for research purposes. For more information, follow the link of the collection.
The Bentham collection consists of a set of images of a collection of works on law and moral philosophy written by the philosopher Jeremy Bentham.
The Hattem collection is a Dutch document that belongs to the "Artes Liberales" literature, from 15th century and belongs to the medical domain.
The Plantas collection is a series of 7 books written by Bernardo de Cienfuegos in the XVII century. It is written in Spanish and it also contains many drawings.
The Reichsgericht collection is a selection of court decisions from the German High Court (Reichsgericht) from 1900-1914.
The Wiensanktulrich collection is a selection of documents with birth records from 16th century written in Wien.


Roberto Rossellini's Generale della Rovere, 1959, while not properly part of his great WW II trilogy which defined the emerging Italian neo-realism (Rome, Open City, 1945, Paisan, 1946 and Germany Year Zero, 1948) continues and extends his use of the war time experience.
When the body contends with the soul for survival, which will win?
Filmed primarily on Rome's Cine Città rather than in street locations as were the others, and using a more professional cast of actors than the nonprofessionals of the earlier trilogy, it nevertheless uses black and white shots of street corners, cafes and Nazi offices to explore choices and survival in occupied Genoa.
Bardone, played by Vittorio De Sica is handsome and graying, an irascible gambler and hustler in Genoa in 1943. The Allies have landed in Naples but Rome and Genoa are still Nazi controlled. He has maintained himself by keeping up affable relations with the occupying Germans and carrying bribes between Italians, frantic to keep sons and husbands off the work trains carrying them to Germany, and the German military bureaucrats with the appropriate files and lists. He also has a gambling obsession, and not a lucky one. Attempting to turn bribe money into better money, he inevitably loses, forcing him rush out to hustle another mark to make up for the losses in the preceding.
The film opens with brief scenes of uniformed Italians singing a fascist song and women in a small piazza thankful that the overcast will prevent bombing raids for the day. We don't have specific details yet but the general tone is set. Into this walks a somewhat furtive man in a black hat and trench coat. A graffitti on a wall "We will win" tells us there is opposition to the earlier fascist sequence. We don't yet know why the man is in a hurry, why he looks worried.
Suddenly he, and we, are startled by the sound of a gunshot. It turns out to be a tire blowing out; a jeep with Nazi soldiers. After some loud commands from the soldiers, the officer, Colonel Mueller [Hannes Messemer], intervenes. Bardone turns on the charm and the two men understand each other; two gentlemen with similar courtesies. Bardone directs them to a tire repair shop; more pleasantries are exchanged, Bardone throws a casual Nazi salute. We don't know him yet and are suspicious of his friendliness to a Nazi; where is this story going? We soon learn that it is through such connections that he makes his living. The Colonel and Bardone have begun a dance to take us through the film; who is leading and who is being led?
This is followed by a scene between Bardone and his mistress, which at first suggests he is able to keep her in comfortable conditions, even though it is war time. The electricity is out, however, and we learn his true situation. He has lost yesterday's bribe money before getting it to the right hands and needs her to pawn some of her jewelry to help him through his "temporary" set back. She icily suggests he pawn the emerald setting he gave her, which he knew when he gave it was a fake. We have the measure of the man in a wonderful two, short scenes.
The middle of the film follows him through more of his dealings and cons and pleadings. On the one hand he is a decent man concerned with the situation of others, with a genuine feeling of wanting to help them. On the other, he is unable to escape his need to be a "player," to be a man of status and respect. Revealing scenes in a bordello, where he is old friends (of course) with the madam and one of the girls, add color and texture to his character. He is trying to peddle the fake emerald– but not to friends. Partisan fighting enters the story. The allies are coming up the Italian boot and irregulars are operating in Nazi occupied northern Italy. An important General, Della Rovere, is killed during an attempt to capture him, to the great displeasure of Colonel Mueller and his superiors. A group of partisans are arrested. Among them is a renowned leader, The Wolf, whom the Germans would dearly like to identify, execute and dishearten the partisans. Mueler concocts a plan to send Bardone into the prison, falsely identified as the dead General, at first to keep news of the General's death quiet, but quickly converted to a scheme to discover which of the imprisoned guerrillas is The Wolf.
The emotional tension begins to build. Will Bardone/Della Rovere, who is being very well treated in the prison, betray a fellow countryman and keep his benefits, or will he keep his silence and possibly face execution? For a movie set prison, it is pretty convincing: rows of cells, doors slamming. Nazi oversight and torture to get information are juxtaposed to Italian conviviality in the cells and between prisoners and Italian guards; the moral dimensions of his choice are drawn in black and white. Though there are scenes that veer towards war-time propaganda characterization of the inevitability of good (our side) overcoming evil (their side) the opposition is nevertheless effective. The closing scene, decision made, is a stark reminder of the choices forced in war time.
A good score by Rossellini's younger brother, Renzo, who also contributed to Open City and Paisan, adds to the emotional effects on the viewers — drum beat danger, ominous clarinet warnings, tension-building string staccato.
Though it is nice to see General Della Rovere in the context of Rossellini's earlier War Trilogy, it stands on its own quite well, a good way to ask ourselves: faced with backing down or facing up, how do we ourselves fare?


Harkening back to the Hollywood's Golden Age, this style delivers a vintage look that is no longer just for the rich and famous.
Allow your beard to grow for three to four weeks.
Shave the hair on the upper parts of the cheeks, leaving a line of hair that extends from your ear to the side of your chin. The beard should not connect to the sideburns.
With your Wahl® trimmer, trim hair clean on the neck; however, you want to leave enough around the chin and jawline to create definition.
Leave two vertical strips of hair connecting the chin with the mustache area.
Grab a trimmer and trim the soul patch to create a perfectly rectangular shape.
Continue to trim the beard as time goes on.


Guilmon Comes Alive (Episode 106): Takato and his friends play the Digimon card game and they all enjoy it. One morning, after Takato and his friends played the game, Takato found a mysterious card, a blue card in his card box. Takato is late for school, so he have to sit outside the class waiting. While waiting, Takato decides to sketch his own Digimon character. Later, the blue card turns itself into a digivice. The digivice then reads the sketch and informations of all the papers that Takato sketched on. A new Digimon was born!
To Fight Or Not To Fight (Episode 108): Rika tells Renamon to attack Guilmon, but after all the direct hits, Guilmon seems to be okay. Guilmon is winning to battle, but stoped after Takato told him to start to listening. Takato and Guilmon united, but Renamon still tries to attack them. Then Henry showed up and stoped Renamon. Rika and Renamon left after Henry arrived; Henry and Takato talk to each other then they went home. The next day when Takato was on his way to visits Guilmon and then he saw government's trucks and got worried. When Takato and Guilmon walk on the street, people asked about Guilmon and Takato told them that Guilmon is a costom he made and his friend is inside the costum. The digivice is beeping and the screen turns red, suddently, Guilmon ran to a packing lot. They met Rika then Guilmon and Renamon starting to fight again! Henry showed up with Terriermon and Terriermon digivolve into his champion form. Gargomon went out of control and tried to hurt Rika, but Takato told Guilmon to stop Gargomon just in time to save Rika.
It Came From The Other Side (Episode 109): With Terriermon as Gargomon, Henry cannot bring his digimon home. Takato is wondering what would Guilmon digivolve into, then decided to draw Growthmon, Guilmon's champion form. The next day when Takato and Henry met each other, they saw a white digital field and started to run away from it. The digital field chase Henry and Terriermon to a big building then Gorillamon appears before their eyes with a desire to fight. Henry remembered his past before he got Terriermon and remembered how it is his own fault. With the help of Takato and Guilmon, Terriermon defeat Gorillamon. Terriermon is about to absorb Gorillamon's data, but Henry told Terriermon that he does not want Terriermon to do that and hoped Gorillamon is back to where it came from.


Cockburn, Alistair. Writing Effective Use Cases. Boston, MA: Addison-Wesley Professional, 2000.
Acceptance Test-Driven Development. Example. This blog post elaborates on using Use Case format to augment a User Story formatted requirement.
Requirements have been written using many formats. In most cases the "User Story" format is best. This article describes how to write in the "User Story" format. It also offers guidelines for choosing another format.
Everyone who writes requirements needs to choose the best requirement format.
The "User Story" format can be used to capture a single scenario, an example of some behavior of the feature, or even reminders or triggers to have a future conversation.
Choosing the best format for a requirement is based on judgment. Judgment comes from experience with requirements decomposition as well as having an understanding of what kinds of information will be needed by those implementing the next level of detail (either requirements at the next level of decomposition, or the software or human procedures that implement a team-level requirement).
Start by assuming the User Story format will work and try to write the requirement using it. If the User Story format does not adequately capture and communicate the requirement, consider alternate formats.
If the User Story format is not capturing every important aspect of the requirement, someone with experience at the next level of detail (or the one below it) should be consulted to find out what they will need to know to do their work successfully. Then pick or develop a format to replace or supplement it.
Tools do not usually influence which requirements format to use. However, tools can help keep visible the context of a given requirement. This in turn can suggest what information needs to be recorded in the requirement. Consider a mind-mapping tool or a drawing tool that can be used to draw flow, System-of-System, and other relationship diagrams.
Parent and Related Stories. Where did the story come from and what else is it related to?
The Use Case is one of the main alternative formats used for requirements.
The Use Case is best for when more detailed information is needed about how the stakeholders will need to interact with the functionality being added to the system. This can include things like the before and after conditions of the system fulfilling the requirement, alternative and essential actions based on differing conditions when the user invokes the functionality in the requirement, etc.
Use Cases can be used at any level, including capabilities expressing Business or stakeholder goals (these are sometimes called "Business Use Cases").
The main benefit of the Use Case format is as a list of additional information to add, to augment a User Story formatted requirement.
The main danger in the Use Case format is in making implementation decisions earlier in the development process than they are needed. For instance, it is tempting to specify the steps of interaction between the primary actor and the system in too much detail, before enough is known about the project to allow choreographing those steps for best effect.
Output from this practice is requirements about which every aspect of importance is expressed, but which contain no unnecessary information such as premature design decisions.
At every level of requirements decomposition.


COR Safety Solutions makes passing COR, MECOR, and SECOR safety audits easy. We provide small and medium-sized companies with all the safety requirements necessary to work Canada's oil patch with none of the hassles that are a natural part of complex compliance systems.
We provide a bridge connecting large oil field production companies and the small and medium contractors who work for them. On one side, we audit companies large and small and on the other we coach and guide contractors through the COR, MECOR and SECOR,and ISNetworld® submission process.
Making Safety and Safety Audits Easy!
COR Safety Solutions will supply certified auditors to audit your company in order to obtain your COR certification. A health and safety audit is a comprehensive review of a company's health and safety system. It covers the basic elements of a health and safety system using personal interviews, a review of documentation and observing operations at the work place. To acquire a COR, you must receive and 80 per cent overall mark with no marks below 50 per cent in any one element of the system.
COR Safety Solutions evaluates your safety program, implements solutions and prepares your company for auditing.
The cost of COR Safety Solutions' services can be customized to you and your company's unique circumstances.
Possibly the most challenging project for small and medium-sized contractors just starting out in business is compiling a safety manual to meet ENFORM requirements for a COR, MECOR and SECOR.
COR Safety Solutions can assist you in creating and implementing a Health and Safety Manual. If you have already compiled your own quality system manual, COR safety manual, SECOR safety manual or other safety or quality systems manual, we can review your documentation prior to your submitting it for certification and provide advice on how it may be improved to assure acceptance.
Alberta Construction Safety Association's Certificate of Recognition (COR) program and SECOR program.
Additional sections can be added.
COR Safety Solutions provides Safety Forms binders, which contain all the forms and paperwork required to stay in compliance with the program — safety meetings, hazard assessments, inspections, etc.
Audits by a certified auditor are required yearly to qualify for and maintain a Certificate of Recognition (COR). COR Safety Solutions can assist your company through each required step ensuring your preparation for each annual internal audit and each external audit, required every three years.
These audits are required when a significant event occurs resulting in injury, property damage and loss of production. We have certified auditors on our team of safety specialists that can audit your program or assess your readiness for an audit. Audits are generally carried out by an external auditor to ensure objectivity.
When there is an increase in the number or severity of incidents.
When there is a change in operational activity or staffing.
When a previous audit has identified a high-risk workplace.
When a certifying partner requires an audit to meet and/or maintain COR certification.
For the Saskatchewan COR program.
When a regulatory agency requests that the entire program is evaluated.
When a company initiates a system-wide audit on its own.
We assess your RAVS® Safety Program Matrix and determine exactly which written programs are required by your customer.
We upload the Adobe PDF format directly to ISNetworld's® website. A safety consultant then completes the questionnaires required for each individual safety program RAVS® section. This step alone will save you many hours of tedious work.
If any program receives less than 100 percent passing, we will make any and all required changes at no additional cost to you.
Once all written safety programs have received a 100 percent passing score, we will email you the RAVS® information that was submitted.
ComplyWorks Ltd. provides relevant compliance with active monitoring solutions in a variety of global industries. These include oil and gas, transportation, construction and forestry.
We focus on three levels of business – the employer, the contractor and the worker. We provide each level with the tools needed to set, manage, monitor, and communicate their compliance in every aspect of their business.
Filling out ComplyWorks questionnaires can be intimidating and time-consuming. We will assist you in submitting this information and guide you through the online submission process.
Please call us to discuss the very reasonable cost of Comply Works Compliance and Comply Works Account Management.


text:
FARRRS road scheme starts topping out process
Another important milestone has been reached on the Finningley and Rossington Regeneration Route Scheme (FARRRS) with construction of the road starting on top of the recently completed eight metre high embankment between Junction 3 of the M18 and Rossington.
Tarmacking of the first section of dual carriageway will provide access into Verdion's £400million iPort project and Harworth Estates' development of 1,200 new homes on the former Rossington Colliery site. The ground works for those developments have already started and building work, including the first homes, is expected to start in spring next year.
As well as enabling work to progress on these schemes, FARRRS will dramatically improve journey times to Doncaster Sheffield Airport supporting development at the new Enterprise Zone and its other business parks.
Excellent progress is being made along the entire three mile route from the motorway network to Parrott's Corner at the A638 Bawtry Road and the link to Rossington at West End Lane.
Embankments continue to go up either side of the new bridge being constructed over the East Coast Main Line and following completion of bridges over the Brancliffe to Kirk Sandall Line and the River Torne at Rossington, work is well underway on the remaining three bridges. Beams are expected to be lifted into place shortly on the Mother Drain bridge on the approach to Parrott's Corner.
Ros Jones, Mayor of Doncaster, said: "We are entering the home straight for the completion of access into private sector led developments which will unlock thousands of new jobs and homes.
"FARRRS is one of a suite of initiatives strengthening our economy and making Doncaster an attractive proposition for new businesses, investors and developers. We have a new Enterprise Zone, the High Speed Rail College, the recently announced Inward Investment Incentive Pilot Scheme and ongoing regeneration schemes which are helping to transform the town. I am fully committed to delivering economic growth in Doncaster."
Tim Love, Director of Strategic Land at Harworth Estates, said: "We saw for ourselves the fantastic work being made on the road a couple of weeks ago and its extremely heartening that the entire programme – one of the most complex in the UK – remains on track. The first homes will be built on our site next March and we look forward to delivering a development that local people will be proud of."
John Clements, Development Director of Verdion, said: "We are extremely pleased that FARRRS is on track to deliver the first stretch of new road which will open up access to iPort. We are now on site starting the £21 million infrastructure works, making the site oven ready for potential occupiers."
As well as assistance from the Government's Regional Growth Fund, the Doncaster Council led regeneration project has backing from private sector partners Harworth Estates, the Peel Group and Verdion and is supported regionally by the Growing Places Fund and the Sheffield City Region Local Enterprise Partnership (LEP). The main contractor for the works is Carillion.
Further information on FARRRS is available at: www.doncaster.gov.uk/FARRRS

redpajama_set_name: RedPajamaCommonCrawl


Arabian Nite Guest House is one of the highly recommended Piet Retief guest houses and provides accommodation in 6 guest rooms for tourists visiting the Grass & Wetlands region.
Arabian Nite is an up market Halaal guesthouse situated among the beautiful forests of Mpumalanga in the little historic town of Piet Retief, a serene haven where the East indeed meets the rest of Africa.
Situated approximately two km outside Piet Retief town on your way to St Lucia and 500m from the Piet Retief hospital you are assured of a home away from home hospitable accommodation.
We currently have six luxurious en-suite fully airconditioned rooms, four rooms have King-size beds and two have Queen-size beds. The guesthouse provides accommodation for up to 18 people.
Each room at Arabian Nite is tastefully decorated and consists of single beds that can instantly be transformed into luxurious King or Queen sized beds therefore catering for everyone in the family. Each en-suite has its own TV and airconditioner as well as kettle along with complimentary tea and coffee.
The spacious breakfast area serves breakfasts for all tastes and special needs (e.g. diabetics and health conscious). Each meal is prepared by Mrs Aboobaker herself ensuring that utmost care and love is put into every aspect of your stay at Arabian Nite.
Our niche is specifically catering for the eastern cuisine lover with a Mediterranean flair, with splendid, awesome breakfasts and dinners for the discerning guest. Lunch packs and dinner are available on prior arrangement with the host.
• 2 hotel Deluxe beds in each of the 6 rooms, which can instantly be converted to king size beds (new concept).
• Spacious bath rooms with showers.
• TV's and complimentary tea and coffee in all rooms.
• Spacious, air conditioned breakfast area, with fridge and microwave.
• Short and long term accommodation.
• Iron and ironing board available on request.
• Baby cot available on request.
There is ample safe and secure parking at Arabian Nite Guest House ensuring that all families are safe and well taken care of.
For those who would like to stay on at Arabian Nite and tour the beautiful Highveld of Mpumalanga, Arabian Nite offers guided tours of the Heritage Route.
We have four luxurious en-suite fully air-conditioned rooms with King-size beds. Each room at Arabian Nite is tastefully decorated. Each en-suite has its own TV and airconditioner as well as kettle along with complimentary tea and coffee.
We have two luxurious en-suite fully air-conditioned rooms with Queen-size beds. Each room at Arabian Nite is tastefully decorated. Each en-suite has its own TV and airconditioner as well as kettle along with complimentary tea and coffee.
Very Peaceful, comfortable rooms. Just Like Home.
Child policyChildren of all ages are welcome. Cots can be provided, if required.


text:
FactoryBot.define do
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    association :service, :all_tiers
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redpajama_set_name: RedPajamaGithub


text:
Volta kan syfta på:

 Elektriska AB Volta – ett svenskt företag som tillverkade dammsugare
 AS Volta – ett estniskt elektromekaniskt företag
 Volta (dans) – en dans
 Volta (musikalbum) – ett musikalbum av sångerskan Björk
 8208 Volta – en asteroid i huvudbältet

Personer
 Alessandro Volta, (1745–1827) uppfinnare av det första elektriska batteriet

Geografi

Belarus

 Volta (vattendrag i Belarus), Vitsebsks voblast,

Ghana

 Voltaregionen, region,
 Volta (flod),

Robotskapade Belarusförgreningar
Robotskapade Ghanaförgreningar

redpajama_set_name: RedPajamaWikipedia


text:
Малое Бабье — деревня в Окуловском муниципальном районе Новгородской области, относится к Боровёнковскому сельскому поселению.

Деревня расположена на Валдайской возвышенности, в 22 км к северо-западу от Окуловки (29 км по автомобильной дороге), до административного центра сельского поселения — посёлка Боровёнка 7 км (14 км по автомобильной дороге).

В окру́ге расположены ещё две деревни: Поджарье — в 2 км к западу и Большое Бабье — в 1 км к северу. В 3 км к востоку ранее существовала ещё одна деревня — Казань.

Транспорт
Ближайшие железнодорожные станции расположены на главном ходу Октябрьской железной дороги в 7 км северо-западнее в посёлке Торбино и в 7 км юго-восточнее в посёлке Боровёнка, а остановочный пункт электропоездов в 2 км к западу от деревни в Вялке.

Примечания

Населённые пункты Окуловского района