# Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

2answers
161 views

### Measurable Functions

How do we prove that a function $f$ is measurable if and only if $\arctan(f)$ is measurable? If I use the definition of measurable functions, that is, a function is measurable if and only if its ...
1answer
65 views

### integrals and characteristic functions

$f$ is Lebesgue integrable over $A$, and $B$ is a measurable subset of $A$. I want to show $$\int_B f=\int_Af\chi_B$$, where $\chi_B$ is the characteristic function of $B$ (it is 1 on B and 0 ...
1answer
277 views

### Lebesgue Convergence using The General Lebesgue Dominated Convergence Theorem

Let ${f_n}$ be a sequence of integrable functions on E for which $f_n \to f$ a.e. on E and f is integrable over E. Show that $\int_E |f-f_n| \to 0$ if and only if \$\lim_{ n\to\infty} \int_E |f_n| = ...
2answers
180 views

### The General Lebesgue Integral

For a measurable function, $f$, on $[1, \infty)$ which is bounded on bounded sets, define $a_n = \int_n^{n+1} f$ for each natural number $n$. Is it true that $f$ is integrable over $[1, \infty)$ if ...
4answers
139 views

### Measure and Lebesgue Integral

I got this exercise as homework and I found some problems in solving it. So I hope that someone can help me. Let $f:[0,1] \rightarrow R$ Lebesgue measurable and $S=\{x \in [0,1]:f(x) \in Z\}$. Show ...
1answer
321 views

### {$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$

Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$. I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$. My attempts: I've found ...
1answer
61 views

### Uniform integrablity of measurable functions

How can I show that if family of $f$ is uniformly integrable then so is {$|f|$}? $($by uniformly integrablity: $\forall \epsilon>0 \ \exists \delta>0: |\int_Ef|<\epsilon,\mu(E)<\delta)$ ...
1answer
94 views

### Limit of a measurable function and the Lebesgue integral

Suppose $\{f_n\}$ is a sequence of lebesgue measurable functions such that $f_n\rightarrow f$, except on a set of measure $0$, as $n\rightarrow\infty$, and $|f_n(x)|\leq g(x)$, where $g$ is ...
2answers
99 views

### Trying to show that a function is zero almost everywhere given a constraint on its Lebesgue measure.

We have that $g$ is a measurable and bounded function on $[a,b]$. I have $\int_a^cg=0$ for every $c\in[a,b]$. I want to show $g=0$ on $[a,b]$ except possibly on a subset of measure zero. Proof. By ...
2answers
94 views

### Limit and Lebesgue integral in a compact

I have problem with the exercise that follows. Let $(z_m)_m \in R^n$ so that $\Vert z_m \Vert \rightarrow \infty$ when $m\to \infty$. Let $f:R^n \rightarrow [-\infty;+\infty]$ integrable. Show ...
1answer
319 views

### Finding Lebesgue Integral of $\frac{1}{\sqrt{x}}$ over $(0,1]$

How do I rigorously discover what $$\int_{(0,1]} \frac{1}{x^{1/2}} = \underset{0 \le \phi \le \frac{1}{\sqrt{x}}}{\sup} \int_{(0,1]} \phi$$ (for $\phi$ a simple function) is? Note that I have ...
0answers
70 views

### $\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm$

Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that $$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$ Proposition: $f$ is ...
2answers
582 views

### Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$

I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a ...
1answer
165 views

### Extension of Fatou's lemma

let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $X$. I want to show if (1) holds, then (2) holds too. \lim_{n \rightarrow ...
1answer
158 views

### Lebesgue Integral on a set of measure zero

I need to show that if $f$ is an integrable function on $X$ and $\mu(E)=0 ,\ E\subset X$; then $\int _E f(x) d\mu(x)=0$ . In my attempts I've showed that \$\forall \epsilon > 0 \ \ \exists ...
1answer
96 views

### using sup of an unbounded function

Is what I'm doing valid if we don't have any information on boundedness of $f$ or $f_n$? let $X$ be a finite measure space and $\{f_n\}$ be a sequence of nonnegative integrable functions, \$f_n ...
1answer
108 views

### Uniform integrability and Lebesgue convergence

A). Given that $|X_n| \leq Y$ and $Y \in L$. Try to show $X_n$ is lebesgue integrable. b). Try to give any example for which $X_n \longrightarrow^{L} X$ yet $\not\exists Y \in L$ with \$|X_n| \leq ...
1answer
246 views

### Integration by parts and Lebesgue-Stieltjes integrals

I want to use Integration by parts for general Lebesgue-Stieltjes integrals. The following theorem can be found in the literature: Theorem: If $F$ and $G$ are right-continuous and non-decreasing ...
3answers
165 views

### Non-Lebesgue Integrability of $1/|x|$ over $[1, \infty)$

How does one show that $\int_\mathbb{[1, \infty)}1/|x|$ is not (Lebesgue) integrable? What I could think of is as follows: Letting $f(x)=1/|x|$ (defined for $|x|\geq 1$), define \$f_n(x)=f\chi_{[1, ...
1answer
68 views

### What can we tell about a sequence of measurable functions on a finite measure space such that $\sup_n \int_X |f_n(x)|^2 d\mu < \infty$?

I found this on a qualifier exam, and I think it will help me understand $L^p$ spaces better. Let $f_n$ be a sequence of measurable function on a finite measure space. Suppose that \sup_n \int_X ...
1answer
108 views

### to show a function is Lebesgue integral

I need to show that $f=\frac{1}{\sqrt x}$ is Lebesgue integrable on [0,1]. My attempt: I need to show $\sum_{m=n}^\infty \frac{m}{n} \mu(E_m^{(n)})$ converges absolutely $\forall n$. ...
1answer
71 views

### Convergence of functions in $L^1$

I am trying to prove a theorem, and I have been able to reduce it to the following question. I feel that this should be easy, but I can't see the solution. If $(g_n)_{n\geq 1}$ is a sequence of ...
1answer
266 views

### improper Riemann integral and Lebesgue integral

Let $f$ be a continuous function on $(0,1]$ and is defined as $f: [0,1] \to \mathbb R$. Show that if $f$ is lebesgue integrable on $[0,1]$, the improper Riemann integral \$\lim_{\epsilon \to 0} ...
1answer
84 views

### What does Luzin's theorem imply?

Luzin's theorem states that: let $f:[a,b]\rightarrow R$ be an a.e. finite function, $f$ is measurable iff $\forall \epsilon \geq 0: \exists \phi_\epsilon$ continuous on $[a,b]$ and \$\mu\{x: f(x)\neq ...
3answers
121 views

### Lebesgue- integrability of roots and powers of a function

If the powers of a function $f$ are Lebesgue integrable what can we say about the original function? for example if we take $f=\frac{1}{x} on [1, \infty]$, it is not integrable but $f^2$ is. Is there ...
2answers
73 views

### Prove $\int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$

Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $$cX := \{ cx \mid x \in X \}.$$ Then $$\int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$$ Now I can prove this for ...
2answers
600 views

### Generalisation of Dominated Convergence Theorem

Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
3answers
114 views

### Is $C_0^\infty$ dense in $L^p$?

I have a question concerning the Lebesgue spaces: Is $C_0^\infty$ dense in $L^p$ ? And if yes, why? Thanks!
1answer
183 views

### Easy application of the Dominated Convergence Theorem?

I am struggling with an application of the Dominated Convergence Theorem (DCT) which has cropped up a few times in various proofs I have been studying, in particular a proof about approximating ...
2answers
67 views

### Expectation and Lebesgue integration question

How might I show that if a random variable (call it Z) is such that EZ (expectation of Z) is finite (i.e. it is Lebesgue integrable), then nP(|Z|>n) tends to 0?
2answers
352 views

### Lebesgue measure sigma algebra

Lebesgue measure on sigma algebra, help ........... Which of the following are sigma algebras? reply with justification please. All subsets in rational numbers { {0},{1},{0,1} }in space {0,1} all ...
2answers
182 views

### constructing a sequence of simple functions with Lebesgue measure approaching the riemann integral

Let $\lambda$ denote the Lebesgue measure on the Borel sets of [0,1]. Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous. I know that the Riemann integral $I:=\int_{0}^{1} f(x)dx$ exists. I also know ...
2answers
113 views

### Cantor ternary set problem

Let C be a cantor ternary set If $x,y \in C,$ then obviously $x-y \in [-1,1]$ Conversely I want to prove that if $w \in [-1,1],$ then there exists $x,y \in C$ such that $x-y=w$ How to prove this ...
1answer
80 views

### Relation among $L^{p}(\mathbb{R}^d)$?

Let $L^{p}(\mathbb{R}^d)$ be the linear space consists of $L^p$-integrable functions on $\mathbb{R}^d$ for $1\le p \le \infty$. Are there any relation among these spaces?
2answers
87 views

### Equicontinuous, differentiable continuous problem

Assume that each of {$f_n : [0, 1] \rightarrow R$} is continuously differentiable I know that if {$f_n'$} is uniformly bounded, {$f_n$} is equicontinuous. However, the converse is NOT true. I want ...
1answer
61 views

### Superposition operator in Sobolev spaces

While working on an elliptic problem in $\mathbb{R}^N$, I met an issue that I cannot work out clearly. Assume that we have a continuous function $g \colon \mathbb{R} \to \mathbb{R}$ such that ...
1answer
112 views

### Compact set in all $L_p$, $1\leq p<\infty$

Suppose $X\subseteq L_\infty$ is a compact subset of $L_p$ for all $1\leq p<\infty$. Does this mean that for every $\epsilon>0$ there exists a measurable set $E\subseteq [0,1]$ with ...
0answers
120 views

### Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
0answers
186 views

### Extended Riemann integrability of  a non-negative function implies Lebesgue integrability?

Let $f$ be a bounded function on a finite interval $[a, b]$ of the real line. If $f$ is Riemann integrable, we denote its Riemann integral by $\mathcal{R}(f , [a, b])$. It is well known that $f$ is ...
1answer
158 views

### Yet another definition of Lebesgue integral

Let $[a, b]$ be a finite interval of the real line. A partition $P$ of $[a, b]$ is a finite sequence of numbers of the form $a = t_0 < t_1 <\cdots < t_{k-1} < t_k = b$ Let $(X, \mu)$ be ...
1answer
158 views

### Another definition of Lebesgue integral

Let $(X, \mu)$ be a measure space. Let $X = A_1 \cup\cdots\cup A_k (A_i \cap A_j = \emptyset$ for $i \neq j)$, where each $A_i$ is measurable. We say $\pi = \{A_1,\dots,A_k\}$ is a finite measurable ...
2answers
115 views

### Lebesgue generalizations of Hilbert spaces?

Is an L[p] space a generalization of Hilbert spaces using Lebesgue integration? And if this is the case, is it true that Holder's and Minkowski's Inequalities are generalizations of the ...
2answers
63 views

### Lebesgue integration for $u \in C^{\infty}_c$

Let $u \in C^{\infty}_c(\Bbb{R}^d)$, where $C^{\infty}_c(\Bbb{R}^d)$ is the family of infintly differentiable functions with a compact support. Is $u$ in $L^2(\Bbb{R}^d)$? I think that $u$ is in ...
2answers
235 views

### Application of Radon Nikodym Theorem on Absolutely Continuous Measures

I have the following problem: Show $\beta \ll \eta$ if and only if for every $\epsilon > 0$ there exists a $\delta>0$ such that $\eta(E)<\delta$ implies $\beta(E)<\epsilon$. For the ...
1answer
118 views

### Does $f_n \to 0$ in $L^1(\mathbb R^2)$ imply that $f_{n_k}(x,\cdot)\to 0$ in $L^1(\mathbb R)$ for almost every $x \in \mathbb R$?

I would like to know what you think about this question. It is a "self-posed" question: I formulated it while I was doing an exercise. Suppose you have \$(f_n)_{n\ \in \mathbb N}\subset ...
2answers
91 views

### Convergence in $L^1$ problem.

Problem: Let $f \in L^1(\mathbb{R},~\mu)$, where $\mu$ is the Lebesgue measure. For any $h \in \mathbb{R}$, define $f_h : \mathbb{R} \rightarrow \mathbb{R}$ by $f_h(x) = f(x - h)$. Prove that: ...
1answer
192 views

### A question about integral operator

I have a question: Prove or disprove that: for every $f\in L^{1}\left(\mathbb{R}\right)$, \sup\left\{ { ...
2answers
290 views

### Derivative of step functions

I was reading up on the Lebesgue integral and how it is computed. And since it is a generalization of the Riemann integral in a more theoretic framework, the same fundamental principle holds, only for ...
0answers
153 views

### Lebesgue Integration of Measurable Function

Can I ask a homework question here? Let $f$ be measurable and nonnegative in $\mathbb{R}^n$ Define a radial function $f^*(|x|)=\inf\{t:\lambda(\{x:f(x)>t\})\leq|x|\}$. Show that ...
3answers
145 views

### Lebesgue Integration problem

Can I ask a homework question here? Assume that $f \in L^q(\mathbb R^d)$ for some $q < \infty$ . show that $\mathrm{lim}_{p \to \infty}||f||_p = ||f||_{\infty}$ $p$ conjugate of $q$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 192, "mathjax_display_tex": 5, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9193135499954224, "perplexity_flag": "head"}