# Tagged Questions

Questions relating to measures, measure spaces, Lebesgue integration and the like.

2answers
23 views

### Baire's theorem from a point of view of measure theory

According to Baire's theorem, for each countable collection of open dense subsets of $[0,1]$, their intersection $A$ is dense. Are we able to say something about the Lebegue's measure of $A$? Must it ...
0answers
25 views

### Given $e \in L(0,1)$, why is $\gamma(t)=\int_0^1 G(t,s) e(s)\, ds$ in $C^2[0,1]\cap C^3(0,1)$?

If I have a function $e \in L(0,1)$, why is the function $\gamma$ defined by $$\gamma(t)=\int_0^1 G(t,s) e(s)\, ds$$ an element of $C^2[0,1]\cap C^3(0,1)$, where: $G$ is the Green's function of BVP ...
0answers
16 views

### Question on a third-order boundary value problems

This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems" if u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
0answers
49 views

### For what $p$ is $x^p$ Lebesgue Integrable?

Revising for an exam on Monday any help with the following question would be greatly appreciated; If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1answer
18 views

### $\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$

For $f\in L^{p}$, $p \in [1,\infty)$ we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$ I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
0answers
24 views

### Measurability of multifunction

Let $f:[a,b]\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$. Suppose $f (.,x, u)$ is Lebesgue measurable for each $(x,u)$. Suppose also that $f$ is continuous at $(x, u)$ ...
5answers
252 views

### Examples of properties that hold almost everywhere, but that explicit examples unknown.

In measure theory one makes rigorous the concept of something holding "almost everywhere" or "almost surely", meaning the set on which the property fails has measure zero. I think it is very ...
2answers
23 views

### Two random variable with the same variance and mean

Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely. My partial answer: By the definition of conditional expectation we have ...
0answers
33 views

### Alternative rigorous definition of a surface integral

Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
1answer
41 views

### Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
1answer
57 views

### Measures on all subsets of $\aleph_0$

A theorem of Ulam says: A finite measure $\mu$ defined on all subsets of a set of cardinality $\aleph_1$ must be $0$ for all subsets if it sends every $1$-element subset to $0$. Will this ...
0answers
20 views

### Lebesgue measure of set $M = \{ [x,y] \in \mathbb{R}^2; 2 < x + y < 3; x < y < 3x \}$?

although we can do this by splitting the area four ways and computing four integrals, my book suggests that I try the substitution $u = x + y$ and $v = \frac{y}{x}$. I expressed $x$ and $y$ in ...
0answers
33 views

### Proving that $\bigotimes_{i=1}^n \cal{B}_{X_i} = \cal{B}_{X}$

Theorem: Given separable metric spaces $X_1,\ldots,X_n$ and $X=\prod_{i=1}^n X_i$, where $X$ has the product metric $d(f,g)=\sqrt{d_1 (f(1),g(1))^2 +\cdots + d_n (f(n),g(n))^2}$. Then ...
1answer
25 views

### Inequality between 2p-norm and p-norm for random variables

Recently I was studying something about random matrix theory, and class of sub-guassian / sub-exponential random variables is of interest. In the literature it gave an inequality as following: ...
1answer
44 views

### The semifinite portion of a measure $\mu$

Let $\mu$ be a measure and define $\mu_1$ such that $\mu(E)=\mu_1(E)$ for $\mu(E)$ finite. And for $\mu(E)$ infinite definite $\mu_1$ such that: (i) if $E$ contains finite subsets of arbitrarily ...
4answers
87 views

### Book Suggestions for an Introduction to Measure Theory [duplicate]

Couldn't find this question asked anywhere on the site, so here it is! Do you guys have any recommendations for someone being introduced to measure theory and lebesgue integrals? A mentor has ...
1answer
28 views

### A Measure For The Space of Probability Density Functions

Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
1answer
45 views

### Measurability of an Indexed Product-Measure

If for any fixed $\omega_1$, $P_{\omega_1}$ is a probability measure and $Q_{\omega_1}$ is a stochastic kernel and both are measurable in $\omega_1$, is the indexed product measure ...
3answers
152 views

### why measure theory

I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary ...
1answer
23 views

### Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?

Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius \$t\in ...
1answer
44 views

### Isomorphism Subalgebra

Given, the unit interval $I$ endowed with the Lebesgue measure $\mu$, and let $A$ be the (Boolean) algebra of Jordan measurable subsets of $X$ with respect to $\mu$, (i.e. those sets that satisfying ...
1answer
48 views

### Simplification of an expression

How do I simplify the following expression? $$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$ where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , \$e ...
1answer
28 views

### Are the continuous functions on $G$ dense in $L^{1}(G)$?

If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
1answer
32 views

### Basic question about the definition of an integral on a measure space

Let $(X,\mathcal{B},\mu)$ be a measure space. $\bf{\text{Definition:}}$ For a non-negative measurable function $f$ on $X$, $E\in \mathcal{B}$, $$\int_{E}f d\mu := \text{inf}\int_{E}\varphi d\mu$$ ...
1answer
77 views

### If $f$ is a bounded measurable function $\Longrightarrow$ there is a sequence of step functions such that $s_n \longrightarrow f \; a.e$?

If $f:[0,1]\longrightarrow\mathbb{R}$ is a bounded measurable function $\Longrightarrow$ there is a sequence of step functions $\displaystyle s_n=\sum_{j=1}^{p} c_j \cdot \chi _{I_j}$ such that \$s_n ...
1answer
21 views

### Conditional expectation is square-integrable

I am given the following definition: Let $(G_i:i\in I )$ be a countable family of disjoint events, whose union is the probability space $\Omega$. Let the $\sigma$-algebra generated by these events ...
1answer
32 views

### Show that E is measurable?

Suppose $E_1= [1, 1 \frac12] , E_2 = (2, 2\frac14), E_3 = [3, 3\frac18], E_4 = (4 , 4 \frac{1}{16}) , \dots , E= \bigcup_{n=1}^{\infty}E_n$ i) Show $E$ is measurable ii) Compute $m(E)$ Here is ...
0answers
60 views

### Let $g$ be a bounded measurable function on $[0,1]$.

Let $g$ be a bounded measurable function on $[0,1]$. For each $n$ Let $\displaystyle I_j=j\cdot \frac{1}{2^n}+[0,\frac{1}{2^n}]$ , $j=0,1\cdots ,2^n-1$ , a partition of $[0,1]$ by bisections ...
1answer
38 views

### A two-dimensional set of measure zero

I have a 2D domain $[0,1]\times[0,1]$. This domain contains some set of measure zero $A$, the last understood as the Lebesgue measure in $\mathbb{R}^{2}$. Is the following true: for almost all ...
1answer
67 views

### How to understand C(X)'' = bounded Borel measurable functions?

Let $X$ be a compact metric space and $C(X)=\{ f:X\rightarrow \mathbb{R} \ | \ \ f \ continuous\}$ with the uniform norm. It is a separable Banach space. 1) I'm aware of the fact that $C(X)^*$, the ...
1answer
48 views

### Riemann integral with intervals?

Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$ Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$. You can use the definition of Riemann integral ...
1answer
35 views

### Counterexample to upper continuity

Let $M$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu:M\rightarrow[0,\infty)$ be a finitely additive set function. I'm trying to decide if it's automatically true that for all ascending ...
0answers
38 views

### Which definition is correct?

I have encountered several different definitions of left Haar measure that don't all seem to agree. The setting I care about is Locally Compact Groups. The first seems to completely disagree with ...
1answer
56 views

### E measurable with m(E) < $\infty$?

Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$. ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable. I told my ...
1answer
35 views

### Haar measure $\tau$-additive?

I'm reading some results from Measure Theory Volume 4 by D.H. Fremlin, and I'm stuck on something. This is pulled out of one of his lemmas (stated more generally for topological groups): A Haar ...
1answer
74 views

### If $f :\mathbb{R}\to\mathbb{R}$ is measurable, then $E = \{x: f(x) \geq 3\}$ is measurable

Prove: Suppose $f : \mathbb{R}\to\mathbb{R}$ where $f$ is measurable and $E = \{x: f(x) \geq 3\}$. Show $E$ is measurable. I saw this statement while reading in a paper and thought this might ...
0answers
24 views

### Lebesgue inner measure

the definition of inner measure: m ∗ (A)=sup{m(S):S∈M,S⊆A} I need to prove: 1) If inner measure=outer measure then A is measurable set 2) m*(AUC)+m*(A n C)>m*(A)+ m* (C) 3)m*(UA)> sum (m*(A)) for ...
1answer
32 views

### Why are Haar measures finite on compact sets?

I'm working through the answer by t.b. to another user's question here: A net version of dominated convergence? because I am trying to work through a related problem and I think it will be ...
0answers
20 views

### Are Haar measures complete?

If $G$ is a locally compact group and $\mu$ is a left Haar measure for $G$, then is the measure space $(G,B(G),\mu)$ complete (where $B(G)$ is the set of Borel subsets of $G$)? Or do we have to take ...
0answers
44 views

### Proving that $\int \left| f-g \right|~d\mu = 2\int_{A_0} (f-g)~d\mu$

Given a (dominant) measure $\mu$, consider two probability measures $f~d\mu$ and $g~d\mu$ over $(\Omega, \mathcal F)$, I'd like to check the following reasoning for showing that  \int \left| f-g ...
1answer
24 views

### Showing that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}[\exp[tX]]$

The problem is to show that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}(\exp[tX])$ given $\exp(tX)<\infty$ for $t\in \mathbb{R}$ where $X$ is a random variable. Then to show that ...
1answer
39 views

### Finitely additive measure on $\mathbb R$

Suppose $\mathcal B$ is the Borel $\sigma$-algebra on $\mathbb R$. Let $\mu : \mathcal B \rightarrow [0, \infty ]$ be a finitely additive(but not necessarily countably additive), ...
0answers
15 views

### Convergence of an Integral in a locally compact group

I'm trying to finish an exercise which I asked about earlier here: Mapping $G$ into its group algebra as left multiplication. Continuous? $\bf{\text{The setting:}}$ Let $G$ be a locally compact ...
0answers
31 views

### question about essential supremum

Consider $u : \Omega \rightarrow \mathbb{R}$ a measurable, nonnegative and bounded function. With $\Omega \subset \mathbb{R}^n$ bounded and open . Is true that \$\mathrm{ess } \inf \ u = \inf \ ...
1answer
29 views

### $\int_\Omega fd\mu_n\to\int_\Omega fd\mu,\ \forall\ f\in C_0(\Omega)$ implies $\mu_n(\Omega)\to \mu(\Omega)$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded open smooth domain and $C_0(\Omega)$ the set of bounded continuous functions with compact support. It is know that $C_0(\Omega)^\star =M(\Omega)$, where ...
0answers
25 views

### Lebesgue Integral Rudin Problem [duplicate]

Suppose {$n_k$} is an increasing sequence of positive integers and E is the set of all x$\in$($-\pi, \pi$) at which {sin$n_k x$} converges. Prove that $m(E)=0$. Hint: For every A $\subset$ E, ...
1answer
36 views

### Calculate an integral in a measurable space

Let $(X,\mathcal{M})$ a measurable set with measure $\mu$. Let $f$ be an integrable non negative function, such that $K:=\int_{E}f \mathrm d\mu<\infty$, where $E\in(X,\mathcal M)$. Let ...
1answer
58 views

### About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
0answers
35 views

### A few questions about Measure Algebras

I've written up some of my understanding as well as I can of the Measure Algebra, trying to see the details behind a very brief treatment. There a couple places where I cannot make see how to make ...
1answer
67 views

### Involution in $L^{1}(G)$ is isometric.

(Sorry for asking so many questions of the same type. There is an underlying issue that I think once resolved will allow me to understand them all at once.) Let $G$ be a locally compact group, and ...
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