Questions tagged [lebesgue-integral]

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

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54 views

$\lim_{n\to \infty}(\int_0^1 f(x)^{2n}g(x)^n h(x)~dx)^{1/n}$ where $f,g,h$ are positive continuous functions on $[0,1]$

I want to find $\lim_{n\to \infty}(\int_0^1 f(x)^{2n}g(x)^n h(x)~dx)^{1/n}$ where $f,g,h$ are positive continuous functions on $[0,1]$. By Holder's inequality, this limit is greater than or equal to $...
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2answers
240 views

Showing that Lebesgue Dominated convergence theorem is false in case of Riemann integration.

I was reading Tom Apostol book called "Mathematical Analysis" and I read this statement: the Lebesgue Dominated convergence theorem is false in case of Riemann integration. Here is the ...
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0answers
25 views

How to calculate 2-dim Lebesgue integral of piecewise function?

\begin{equation} f(x,y)= \begin{cases} 1, & \text{if}\ x>=0 \text{ and } x<=y<x+1\\ -1, & \text{if}\ x>=0 \text{ and } x+1<=y<x+2 \\ 0, & \...
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0answers
34 views

Integrability of $\frac{|xy|^a}{nz^3}$ over $\sqrt{x^2+y^2}<z<\sqrt{n^2-x^2-y^2}$

Let $a>0,$ $$f_a(x,y,z)= \frac{|xy|^a}{nz^3}$$ on $$A_n=\{(x,y,z):\sqrt{x^2+y^2}<z<\sqrt{n^2-x^2-y^2}\}$$ I want to find all $a>0$ such that $f_a \in L^1(A_n)$ for $n \geq1.$ Looking at ...
2
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1answer
67 views

Monotone Convergence theorem Application

$$ \lim _{n \to \infty} \int_{-\infty}^{\infty} \frac{e^{-x^{2} / n}}{1+x^{2}} d x=? $$ My opinion is using Monotone Convergence Theorem here. For every $x \in \mathbb{R}$ the sequence $\left\{e^{-x^{...
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1answer
37 views

Integral Criteria for Functions to be Zero Almost Everywhere

While reading the proof of Lemma 2 in the following link, I realized they only proved the case of a nonnegative function $f$, but that's not an hypothesis of the lemma. So, what happens if $f$ takes ...
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1answer
21 views

An application of the dominated convergence theorem to approximate a function.

In "Measure theory and probability theory (pag. 58)" by Krishna and Soumendra we can found the following: Let $\mu$ be a Lebesgue-Stieltjes measure, and $f \in L^p(\mathbb{R}, \mathcal{B}(\...
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1answer
45 views

Integrability of $f(x,y)=\frac{(x+y)\log(y/x)}{(xy)^{a}(1+x^2+y^2)}$ on $O=\{(x,y):x>0,y>0\}.$

Let $$f_a(x,y)=\frac{(x+y)\log(y/x)}{(xy)^{a}(1+x^2+y^2)}=\frac{1}{x^a}\frac{(x+y)\log(y/x)}{y^a(1+x^2+y^2)} $$ defined on $$O=\{(x,y):x>0,y>0\}.$$ I need to find all $a\in \mathbb{R}$ such that ...
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1answer
36 views

For any pair of sequences $(a_n),(b_n)$ with $a_n<b_n$ and $a_n \rightarrow \infty$ we have $\lim_n \int_{a_n}^{b_n} f d \mu=0$

I wrote down a solution, but I'm not sure if it works: Question: Let $\mu$ be a measure on $\mathscr{B}(\mathbb{R})$ such that $\mu(B)< \infty$ for all bounded $B$. Let $f \geq 0$ be $\mu$-...
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2answers
80 views

Prove that $\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$ [closed]

Let $f:[0,\infty)$ be Lebesgue-integrable, then prove that $$\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$$ This is also known as Cesàro ...
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1answer
30 views

Show that the function $f(x)g(x)$ is integrable(2).

As a completion of this question here Show that the function $f(x)g(x)$ is integrable. I do not know how to answer $(b)$ and $(c)$ below. Let $A:=[a,b].$ Suppose that the function $f: A \rightarrow \...
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1answer
30 views

A question in functional equation of Gamma Function

I have a question in Tom M Apostol ( Mathematical Analysis) Example on Page 278. I beg for every one's pardon, by mistake a different image was added. I am really sorry for wastage of other people's ...
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1answer
49 views

Deducing Lebesgue Integration from Riemann Integrability of a function [duplicate]

I have a question which I thought while studying a theorem in Chaptee Lebesgue Integration from Tom Apostol and it may seem easy but can anyone please tell about it. Question: Assume that a function ...
1
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1answer
32 views

Show $ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff f_n\to f \text{ a.e.}$

Consider $(f_n)_n$ an increasing sequence in $\mathcal{L}^1$ and $f\in \mathcal{L}^1$. Show that $$ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff ...
2
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1answer
49 views

Application of of Lebesgue integral properties in the scope of simple functions

As I continue going through measure theory with Folland and the introductory text by Tao, I came across this simple problem in a problem note online regarding unsigned Lebesgue integrals. Let $(X, F, ...
2
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1answer
68 views

Proving a subset of $H^1(\mathbb{R}^d)$ is compactly embedded in $L^2(\mathbb{R}^d)$.

I was recenty reading about the weighted Lebesgue spaces and came accross an exercise that asks to prove that $H^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d,|x|^2\,dx)$ is compactly embedded in $L^2(\...
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1answer
11 views

if $y,z$ converges to zero, does $yz \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$ as well?

Let $\Omega=(0,1)$ and $y,z \in L^2(\Omega)$ such that $$y \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$$ $$z \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$$ can we deduce that $yz \rightarrow 0 \,\, \...
2
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1answer
36 views

An integrable function $f$ on $\Bbb R$ satisfying $\lim_{h\to 0}\int_\Bbb R \frac{|f(x+h)-f(x)|}{h}dx=0$ must be constant

Suppose $f:\Bbb R\to \Bbb C$ is an integrable function, i.e. $\int_\Bbb R|f|~dx<\infty$, satisfying $$ \displaystyle\lim_{h\to 0}\displaystyle\int_\Bbb R \dfrac{|f(x+h)-f(x)|}{h}dx=0.$$ I am trying ...
1
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1answer
30 views

A question in Proof of Theorem 10.27 of Apostol Mathematical Analysis

While self studying Lebesgue integration from Tom M Apostol I am unable to think about an how to deduce a line in the proof whose image I am adding( line is highlited. I am not able to deduce the ...
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1answer
26 views

A question in proof of Apostol ( Mathematical Analysis) in Theorem 10.27

I am self studying Apostol Mathematical Analysis Chapter->Lebesgue Integration and I was unable to think about an argument used in that proof. Adding it's image -> Can someone please tell a ...
1
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1answer
51 views

Integrability of $f(x,y)=(1-x)^{a}$ on $D=\{(x,y):0<x^2+y^2<1\}$

As it is said in the title, I need to check the integrability of $$f(x,y)=(1-x)^{a}$$ for $a\in \mathbb{R}$ on the open disk $D= \{0<x^2+y^2<1\}.$ My attempt Let $a \geq 0,$ then since $D \...
3
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1answer
93 views

Proof that $f$ is Lebesgue-integrable in $[0,1]$.

Let $f:[0,1]\to\mathbb{R}$ be a non-negative function. For all $\epsilon\in(0,1]\,$, let $f$ be Riemann-integrable in $[\epsilon,1]$. Show that $f\in L_{1}[0,1]\,$ iff $\,\,\lim_{\,\epsilon \to 0}\...
1
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1answer
48 views

Show that the function $f(x)g(x)$ is integrable.

Let $A:=[a,b].$ Suppose that the function $f: A \rightarrow \mathbb{R}$ is continuous, $g: A \rightarrow \mathbb{R}$ is integrable and $g(x) \geq 0$ for almost all $x \in A.$ $(a)$ Show that the ...
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1answer
51 views

$f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$ but not converging to $0$ almost everywhere.

I'd like to understand the costruction of the following function we should statisfy the requests of the title. We define $S_{n} := \sum\limits_{k=1}^{n}\frac{1}{k}$, with $a_{n} := S_{n} - \lfloor S_{...
1
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1answer
61 views

How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?

My professor gave us the following form of Dirichlet function as an example of the problems we faced in Riemann integration: $\{r_{n}\}$ enumeration $\mathbb{Q} \cap [0,1]$ $$ f_{n}(x) = \begin{cases} ...
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1answer
21 views

Asking for help in proof of part of a Theorem in Apostol Mathematiccal Analysis ( A part which is not proved)

While self studying Lebesgue Integration from Apostol Mathematical Analysis I am struck on this theorem Image-> I am not able to understand how if theorem is true for {$s_n$ - $s_1 $} then it's ...
2
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0answers
41 views

Asking for proof verification of a theorem in Apostol Mathematical Analysis

I am self studying Apostol Mathematical Analysis Ch-> Lebesgue integration and I have a different thinking about a proof whose only outline is given in Apostol Book. It's image is -> In part (a)...
2
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3answers
117 views

Does Lebesgue integration guaranteed for us that we can **always** integrate after differentiation?

Our professor gave us this function as a problem of Riemann integration to explain why we need Lebesgue integration : $$ f(x) = \begin{cases} x^2 \sin{\frac{1}{x^2}} & if \quad x \neq 0 \\ ...
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0answers
43 views

Unable to think about a step in proof of a theorem in Lebesgue Integration

I am self studying Analysis from Tom M Apostol and I have a question in proof of Theorem 10.2 . I am adding it's image -> The question is in the highlighted line How is A union of intervals that ...
2
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1answer
38 views

How Lebesgue integration solved the problem of changing the order of integration will change the value of integration?

Our professor started a course in measure theory by stating the problems of Riemann integration. One of the problems he\she stated is the following double integration: $\int_{0}^{1}\int_{0}^{1} \frac{...
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1answer
49 views

Where does this equality come?

I need to prove this theorem : Let $X$ be a random variable. Then $|E(X)|<\infty $ if and only if $$\sum_{n=1}^\infty P(|X|>n)<\infty$$ I understand the proof until somewhere but I have ...
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0answers
23 views

Limit of Integral of Continuous Function

I'm trying to solve the following exericize (S.J. Taylor - Introduction to measure and Integration Exercize 5.5.10) The statement to prove is the following: If $f:\left[a,b\right]\rightarrow \mathbb{R}...
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2answers
32 views

If $f\in L^1(\Bbb R)$ is continuous then $f\in C_0(\Bbb R)$

Suppose $f \in L^1(\Bbb R)$ is continuous. Then is it necessarily true that $f\in C_0(\Bbb R)$? It is easily seen to be true if $f$ is uniformly continuous, but I can't see whether it is true if $f$ ...
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1answer
42 views

If $f \in L^p(\mathbb{R})$ then $\lim_{x \rightarrow \infty} \int_x^{\infty} |f(t)|^pdt=0$ [duplicate]

If $f \in L^p(\mathbb{R})$ then $\lim_{x \rightarrow \infty} \int_x^{\infty} |f(t)|^pdt=0$ where $ p \in (1,\infty)$ I tried to use this Problem but if $x\to\infty$, also $a\to\infty$, so $\int_a^{\...
1
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1answer
70 views

Discussing a problem with Riemann integral.

As an introduction to Lebesgue integration, our professor gave us some problems of Riemann integration. One of these problems is the following function: $$f_{n}(x) = n^2 x e^{-nx}.$$ He said the ...
0
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1answer
30 views

An application of Radon-Nicodym theorem

Consider $M$ be the $\sigma -$algebra of Lebesgue measurable sets and $\mu $ the Lebesgue measure. Denote by $P$ the set of $p-$measurable sets, that is the sets $A\in \mathcal{P}\left( \mathbb{R} \...
1
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1answer
29 views

The case of Young's inequality for convolution for $1\leq p\leq 2$ implies the case $p\geq 2$

I'm doing an exercise asking us to show that if for $ 1\leq p\leq 2$, the inequality for convolution $||f*g||_p\leq ||f||_1 ||g||_p$ holds, then it holds for $p\geq 2$. The exercise suggests using ...
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1answer
32 views

Lebesgue integration over domains

Is $\int\limits_{X_1\setminus X}f\mathrm=\int\limits_{X_1}f\mathrm -\int\limits_{X}f$ always true on disjoint interval $X_1\setminus X$ I tried to show from addivity over domains of Integration but I ...
0
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1answer
70 views

$\lim_{\lambda \to \infty}\dfrac{1}{\lambda}\int_0^{\lambda}yf(y)dy = 0$?

Assume that $f : [0,\infty) \to \mathbb{R}$ is a Borel--measurable function. Assume also that is integrable with respect to the Borel measure on $[0,\infty)$. Is it true that: $$\lim_{\lambda\to\infty}...
1
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1answer
34 views

Convergence of sequence of Lebesgue integrals

Suppose we have a sequence of non-negative random variables $\{X_n\}_{n \in \mathbb{N}}$ which are integrable with respect to a probability measure $P$. Denote $\mu_n \equiv E(X_n)$ and suppose that $...
1
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0answers
28 views

IF $f\in L^1[0,1]$ satisfies $\int_0^1 fg^{(n)}dx=0$ for all smooth $g$ supported in $(0,1)$ then $f$ is a polynomial of degree $\leq n-1$.

Let $V$ be the space of infinitely differentiable real-valued functions $[0,1]\to \Bbb R$ supported in $(0,1)$. For some fixed positive integer $n$ and a function $f\in L^1[0,1]$, I am asked to show ...
1
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0answers
39 views

Lebesgue Integral vs Completion of L1

There's a general result which says any metric space can be embedded within a complete metric space(or in other words, the metric space can be completed). If one completes the space of L1 Riemann ...
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1answer
18 views

Example of unequal iterated integral but that does not contradict Fubini's Theorem

Consider counting measure $\mu_1$ and $\mu_2$ on $X=Y=\mathbb{N}$ Define a function, $$ f(x,y) = 2-2^{-x} \ \text{if} \ \ x=y \\ \text{and}\\ f(x,y) = -2 + 2^{-x} \ \text{if} \ \ x=y+1 $$ I showed ...
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0answers
34 views

estimate a integral with parameter

Consider the following integral: $$J(u)=\int_{0}^{u^2}\int_{0}^{u}\frac{1}{x^3+y+3x^2+5x+3}dxdy$$ My question: I want to separate it into two parts: $$J(u)=u^aI(u)$$ where $a\geq 0$ and $I(u)$ is &...
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2answers
27 views

Problem about sequence of non-negative function

Suppose $\{f_n\}_{n\geq 1}$ is a sequence of non-negative functions. define $g_n = \max_{1\leq i \leq n} f_i$ I tried to show that $$ \int_{g_n \geq a} g_n d\mu \leq \sum_{i=1}^n \int_{f_i \geq a} f_i ...
3
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1answer
62 views

Integrability of $\frac{\arctan^a (|x|+|y|)e^z}{n^4+4x^2+y^2} $ on $D=\{(x,y,z):1<x^2+y^2+z<2\}$

Let $$f_n(x,y,z)= \frac{\arctan^a (|x|+|y|)e^z}{n^4+4x^2+y^2} $$ on $$D=\{(x,y,z):1<x^2+y^2+z<2\}$$ I want to determine for which $a\in \mathbb{R}$ we have $f_n\in L^1(D)$ for all $n\geq 1.$ ...
0
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2answers
47 views

Derivate $F$ where $F$ is defined by a Lebesgue integral.

Calculate $F'(x)$ where $F$ is defined by Lebesgue integral $$F(x)= \int_0^1 \frac{sin(xt)}{1+t}dt$$ It is my first time to calculate these kind of derivate, and I don't find exmaples to do it. So ...
2
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2answers
65 views

Showing $\{X_n\}$ is uniformly integrable when $\sup _{n} \mathbb{E}\left[X_{n}^{2}\right]<\infty$

I got a question that show that a family of rvs $\left\{X_{n}\right\}$ is uniformly integrable when $\sup _{n} \mathbb{E}\left[X_{n}^{2}\right]<\infty$ What I have tried: $$\sup_n\mathbb{E}[|X|] =...
1
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1answer
48 views

Understanding the proof of the Dominated Convergence Theorem

I was going through the proof of the Dominated Convergence Theorem. Now if we have that $(f_n)$ is a sequence of measurable functions such that $\lvert f_n\rvert\le g$ for all $n$ where $g$ is ...
0
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1answer
32 views

$\lim_{n\rightarrow\infty}f(x+n)=0$ for almost all $x$ if $f:[0,\infty)\rightarrow[0,\infty)$ is Lebesgue-integrable

I want to show that $$\lim_{n\rightarrow\infty}f(x+n)=0$$ for almost all $x$ if $f:[0,\infty)\rightarrow[0,\infty)$ is Lebesgue-integrable. I attempted a proof by contrapositive, testing for the ...