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.

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2
votes
1answer
47 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
1
vote
0answers
24 views

Are $L1$ functions with a.e. finite support a.e. equal to a continuous function?

I was wondering about this: Let $f \in L^1(\Omega)$ and $\Omega\subset \mathbb{R}^n$ be compact, then $f$ is the $L^1$ limit of continuous functions with support in $\Omega$. Egorov's theorem tells us ...
0
votes
1answer
41 views

prove the equivalence between a null set and a limit

I'm asked to prove that for any non-negative, measurable and integrable function $f$ on $[0,1]$, we have $\lim\limits_{a\to 0}\int_{0}^{a}fdx=0$. I want to use the theorem that for null set E, such a ...
2
votes
1answer
55 views

Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.

Let $g$ be a non-negative measurable function. For $1 \leq p < \infty$, show that $$\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$$ where $\mu$ is the Lebesgue measure and we are ...
2
votes
2answers
40 views

Comparing limits of integrals

If $$f_n:X\rightarrow [0,\infty]$$ is a sequence of measurable functions and we know that $$\lim_{n\rightarrow \infty }\int_X f_n \,d\mu=0,\qquad \qquad \tag{$\star$}$$ then can we conclude that ...
3
votes
1answer
80 views

Limit of the integral $\int_0^1\frac{n\cos x}{1+x^2n^{3/2}}\,dx$

Prove that $\displaystyle\int_0^1\frac{n\cos x}{1+x^2n^{\frac32}}dx\rightarrow0$ as $n\rightarrow\infty$. $f_n(x)=\frac{n\cos x}{1+x^2n\sqrt{n}}$ tends to zero function pointwise. It just ...
6
votes
1answer
40 views

how to prove $\int{f}d\mu=\sum_{x\in\Omega}f(x)$

Prove that $\int{f}d\mu=\sum_{x\in\Omega}f(x)$ when $f$ is absolutely summable, where $\mu$ is a counting measure on the measure space $(\Omega,\mathscr{F})$. Can someone give me hints?
0
votes
1answer
32 views

Definition of Lebesgue integrable function

If a function $f : \mathbb{R}^d \to [-\infty,\infty]$ is Lebesgue integrable, then by definition we have $$\int_{\mathbb{R}^d} |f(x)| \, dx < +\infty.$$ Is it possible to say that there exists a ...
4
votes
2answers
53 views

Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$

Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$. I have seen a proof of this already in a lecture but I ...
6
votes
2answers
98 views

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e.

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e. I let $E \subset \mathbb{R}^d$ be a finite measurable set. I try to break this into two cases: Case 1: If $f(x)=0$ ...
1
vote
1answer
24 views

Dominated convergence and fundamental lemma of the calculus of variation

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
5
votes
1answer
92 views

Trying to calculate the integral limit $\lim_{n\rightarrow\infty} \int_{-\sqrt n}^{\sqrt n}\left (1 - \frac{x^2}{2n}\right)^ndx$

How to calculate following integral: $$\lim_{n\rightarrow\infty}\int_{-\sqrt{n}}^{\sqrt{n}}{\left(1-\frac{x^2}{2n}\right)^n}dx$$ Prove that this integral exists and compute its value. I just ...
1
vote
0answers
22 views

improvement of upper Lebesgue sum

In Pugh's real mathematical analysis, lower and upper Lebesgue sum are given as: $\underline{L}(f,Y)= \sum_\limits{i=1}^{\infty}y_{i-1}\cdot mX_{i-1}$ $\overline{L}(f,Y)= ...
1
vote
1answer
48 views

show Lebesgue dominated convergence theorem fails for ${n^2xe^{-nx}} x\in [0,1]$

show Lebesgue dominated convergence theorem fails for the sequence of functions $f_n=n^2xe^{-nx}$ $x\in [0,1]$ Here is my solution. Is it correct? $f_n$ is an integrable function the sequence ...
2
votes
2answers
37 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwartz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
3
votes
0answers
128 views

Dense subset of $L^p[a,b]$

It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ? ...
2
votes
2answers
48 views

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?
-1
votes
2answers
32 views

Is this subspace of $L^1(\mathbb{R},m)$ closed? [closed]

Let $K$ be the subspace of $L^1(\mathbb{R},m)$ which contains precisely the functions such that $\int f=0$. Is $K$ closed? (EDIT: When I asked this question, I could only see that ${f:||f||_1=0}$ is ...
3
votes
0answers
50 views

Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
1
vote
0answers
19 views

property of distribution function

Let $f$ a continuous map from $\mathbb{R} \rightarrow \mathbb{R}$ and let $L_1, L_2$ 2 probability measures on $\mathbb{R}$. Let $K$ be a closed set in $\mathbb{R}$. In a proof, I want to use the ...
1
vote
2answers
33 views

Application of dominated convergence theorem- find limit

Find (with justification) $$ \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx $$
2
votes
0answers
37 views

What does it mean when $\sigma$ is discriminatory?

I am reviewing this paper on the approximating power of neural networks and I came across a definition that I could not quite understand. The definition reads: where $I_n$ is the $n$-dimensional ...
0
votes
1answer
11 views

Convergence in absence of DCT

Can you give an example of a sequence of non-negative functions tending to zero pointwise such their integral tends to zero but there is no integrable function which bounds them?
1
vote
0answers
32 views

Checking $g(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)\in L_1$ for $r_i\in\mathbb{Q}$

Let $x^{-1/2}$ for $0<x<1$ and $f(x)=0$ otherwise. Let $(r_n)$ be an enumeration of $\mathbb{Q}$ and let $g(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)$. Show that $g\in L_1$ and in particular, $g$ is ...
0
votes
1answer
25 views

Lebesgue Integrable functions

I am in need of guidance for the following question: Let $f:X\to\mathbb{R}$ be an integrable function. Show that $\mu(\{x:|f(x)|\geq n\})\leq 1/n\int |f|\mu(dx)$ for each $n>0$.
2
votes
3answers
62 views

Showing $\int_E f=\lim_{n\to\infty}\int_E f_n$ for all measurable $E$

The following is an exercise from Carothers' Real Analysis: Suppose $f$ and $f_n$ are nonnegative, measurable functions, that $f=\lim_{n\to\infty} f_n$ and that $\int f=\lim_{n\to\infty}\int ...
4
votes
2answers
58 views

Is there a notion of indefinite Lebesgue integral?

When I started studying integration rigorously via the Riemann and Lebesgue integrals, one thing that struck me is that we loose completely the concept of indefinite integrals. Integrals of functions ...
3
votes
1answer
72 views

Lebesgue integral and iterated integral

I am learning lebesgue integral at the moment, and come across a question in homework, but find it really confused. The question states: I first tried to compute the iterated integral by Riemann ...
2
votes
0answers
102 views

upper lebesgue sum with a new partition

Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable.Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is ...
1
vote
0answers
29 views

Prove weak derivative commutes with difference quotient

Let $U$ be an open set in $\mathbb{R}^n$,$f:U\to \mathbb{R},f\in W^{1,p}(U)$. Let $\tau_{h,i}f(x)=\frac{f(x+he_i)-f(x)}{h},h>0$ Given any compact $V\subset U$, show there exists $h_0>0$ such ...
2
votes
1answer
38 views

Lebesgue integral of absolute value of sequence of functions [duplicate]

I am working on a problem$^{(*)}$ on Lebesgue integral looks like this: Given that both $f_n$ and $f$ are integrable, $f_n \longrightarrow f$ a.e., and $\int|f_n| \longrightarrow \int |f|$. Show ...
3
votes
1answer
64 views

$L^2$ and $L^1$ space problem

For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
6
votes
1answer
163 views

Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so it weakly converge to $u\in W^{1,p}_0(\Omega)$ and strongly converge to $u$ in $L^p(\Omega).$ We define a function $f:\Omega\times ...
0
votes
2answers
154 views

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why?

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why and why isn't this true for $1/x$? I'm having difficulty understanding difference between the above functions in terms of ...
1
vote
1answer
41 views

Confused about switching Lebesgue integrals for Riemann integrals

Hi I have been attempting given in the link below. I am confused about the argument used to show the function is not Lebesgue integrable. This question What each person has used to answer is the ...
3
votes
1answer
43 views

Convergence in $L^1_{loc}$ implies convergence almost everywhere

Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ ...
3
votes
2answers
44 views

Proving that $m(E) = 0$ if for all $n$, $\int_E x^n \cos x\, d x = 0$

Suppose that $E\subset [0,2\pi]$ is measurable and $\int_E x^n \cos x\,dx = 0$ for all $n =0,1,2,\cdots$. Then prove that $m(E)=0$. In a non-rigorous fashion, if $\sum_{1}^{\infty} a_nx^n = \sec ...
0
votes
3answers
109 views

Is a compact set an union of a finite number of disjoint closed intervals?

I think it is true for $\mathbb R$ with usual metric. How about others? How to prove it? Motivation: I got this idea when I was reading a proof for Lebesgue's Criterion of Riemann Integrability, here ...
2
votes
0answers
44 views

If a simple function is measurable, then is its characteristic function

I am working on a problem looks like this: If a simple function $s$ is measurable, show that its characteristic function $\mathcal X_{X_i}$ is measurable. Here are the ways I have been working ...
1
vote
2answers
97 views

Lebesgue Integral: $\int_1^{\infty}\frac{1}{x}$

The following is an exercise from Carothers' Real Analysis: Show that $$\int_{1}^{\infty}\frac{1}{x}=\infty$$ (as a Lebesgue Integral). Attempt: Let $E=[1,\infty)$. $\int_E f=\int f\cdot ...
0
votes
0answers
35 views

A question about countably subadditive property of Lebesgue Outer Measure

Here is the definition of Carothers' Lebesgue Outer Measure: . And countably subadditive property of Lebesgue Outer Measure has been talked here: I can understand all proofs. However, I'm ...
0
votes
1answer
38 views

How will m*(rE) behave?

Let $rE =\{rx: x\in E\}$, what is $m^*(rE)$ in terms of $m^*(E)$? Intuitively, I think $m^*(rE)\leq r\times m^*(E)$. However I've no idea how to prove it? Add: Definition of Lebesgue Outer Measure ...
1
vote
1answer
31 views

Disjoint convex sets which cannot be separated by any continuous linear functional

This problem is out of Rudin's Functional analysis exercise 3.2. The problem is stated below. I'm really struggling with this chapter in general. It has a lot of new topics I have not seen before. Any ...
0
votes
0answers
21 views

Conditions for 2 variable functions to be Lebesgue integrable

I'm trying to solve this problem but am having some issues. So I understand the conditions required to show a 1 variable function is integrable on some $E$, a subset of $\mathbb{R}$. But, if the ...
0
votes
0answers
39 views

Integral of limit of a function

I am working on a problem$^{(1)}$ similar to this 2013 posting: Suppose that $f_n$ is a sequence of integrable, non-negative functions, so that $\forall x$, $f_n(x)$ decreases to $f(x)$. Show the ...
0
votes
1answer
61 views

How to explain the why here?

Reading Lebesgue outer measure of Lebesgue Measure Chapter from Carothers' Real Analysis and some properties and their proofs are here: Basically, I can't understand the proof for reverse ...
1
vote
2answers
48 views

On continuous function of compact support

Hello all I am stuck on the following small question in real analysis for practice in which we are given a function f of compact support and a measurable set A we are asked to prove the following is ...
0
votes
2answers
34 views

Countable additivity of Lebesgue integrals proof

Show if $f_n$ are non negative measurable functions: $$\int (\sum_{n=1}^\infty f_n) d\mu = \sum_{n=1}^\infty \int f_n d\mu$$ Does this not just follow from the theorem for two additivity? Say $\int ...
2
votes
1answer
51 views

Showing that $\int_{E\cup F}f=\int_E f+\int _F f$, where $E\cap F=\emptyset$

I would like to show that $\int_{E\cup F}f=\int_E f+\int _F f$, where $E\cap F=\emptyset$ and $E,F$ are Lebesgue measurable sets. Attempt: First I tried to show that in general I can write $\int ...
0
votes
0answers
59 views

Question in real analysis and Lebesgue integration

So yeah my school makes us take a mini real analysis course for physics and I am really stumped on this one: I have the following question and would certainly appreciate any help please : we are given ...