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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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 ...
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17 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 ...
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2answers
22 views

Application of dominated convergence theorem- find limit

Find (with justification) $$ \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx $$
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31 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 ...
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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?
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28 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 ...
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1answer
20 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 $n>0$.
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3answers
47 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 ...
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27 views

Question about Lebesgue integration or lebesgue measure [on hold]

Give me some ideas to solve these problems i read the textbook several times but i can't solve them
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0answers
27 views

Minkowski integral inequality in Banach space

Let $X$ be a Banach space of all measurable functions on $\mathbb{R}^d$ with the property: for any non-negative increasing sequence $\{f_n\}\subset X$, we have $\|\lim_{n\to \infty} f_n\|_X=\lim_{n\to ...
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2answers
51 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 ...
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1answer
50 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 ...
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0answers
72 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 ...
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18 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 ...
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1answer
26 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 ...
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1answer
34 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?
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106 views
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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 ...
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1answer
39 views

How can I solve like this exercise in measure theory [closed]

If $J=\{[a,b[$ : $a \le b$ : $a,b \in R\}$ and $F$ is an continous increasing bounded function on $R$ , and if we put $λ([a,b[)=F(b)-F(a)$ prove that : $$λ(\emptyset) = 0 $$ and if the union of ...
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2answers
118 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 ...
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1answer
35 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 ...
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1answer
41 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 $[a,b]$ ...
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2answers
40 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 ...
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1answer
53 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 ...
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0answers
34 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 ...
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2answers
78 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 ...
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0answers
21 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 ...
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1answer
34 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 ...
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1answer
23 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 ...
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18 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 ...
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0answers
37 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 ...
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1answer
56 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 ...
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2answers
41 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 ...
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2answers
30 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 ...
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1answer
47 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 ...
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0answers
55 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 ...
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1answer
27 views

Question in real analysis an Minkowski difference

I have the following question in real analysis: I was first asked to prove that if the function which I proved to be continuous has a point x such that F(x) > 0 then there exists an open cube in ...
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2answers
46 views

Use the Lebesgue's Monotone Convergence Theorem and the fact that $\int^\infty_0t^ne^{-t}dt=n! \text{ to prove that }\int^\infty_0f(t)e^{-t}dt=s.$

Consider a sequence of real numbers $(a_n$ with $n\in\mathbb{N}_0$, such that $a)n \geq0\text { non-negative}$ for all $n\in\mathbb{N}_0$ and $\sum^\infty_{n=0 } =s\in\mathbb{R}$ b. Use the ...
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0answers
36 views

Question about definition of the Lebesgue integral of a non-negative function

I am reading Royden's Real Analysis to learn about Lebesgue integration. Royden first shows that a bounded function on a set of finite measure is Lebesgue integrable if and only if it is measurable. ...
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1answer
37 views

A rigorous meaning of “induced measure”?

In my readings I often come across terms like "induced measure" or "induced Lebesgue measure". For example: $$\int_{\mathbb{B}^n}u\frac{\partial v}{\partial x_j}\;dx = ...
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3answers
63 views

Integration of $\int_{0}^{1} x^{a}(1-x)^{-1}\log (x) dx $

I need to compute the following integral for $a>-1$, $$\int_{0}^{1} x^{a}(1-x)^{-1}\log (x) dx $$ My attempt: By change of variable $x=1+t$: \begin{align*} I &= \int_{0}^{1} ...
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2answers
62 views

integration of $\int_{0}^{\infty} e^{-ax}\Sigma_{0}^{\infty} \frac{(-1)^n x^{2n}}{4^n (n!)^2}dx $

I need to compute the following integral for $a>1$ $\int_{0}^{\infty} e^{-ax}\Sigma_{0}^{\infty} \frac{(-1)^n x^{2n}}{4^n (n!)^2}dx $ My attempt $\int_{0}^{\infty} e^{-ax}\Sigma_{0}^{\infty} ...
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1answer
29 views

$x^{-1}\sin x$ is not integrable on $[0,\infty)$ implies $e^{-xy}\sin x$ is not integrable on $[0,\infty) \times [0,\infty)$

I am reviewing a homework problem and I came upon the following statement, which is only a part of what I am trying to solve: Given that $\frac{\sin x}{x}$ is not Lebesgue integrable on ...
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1answer
43 views

Integration on manifolds and improper integration

Consider the usual concept of integral on a smooth manifold (the one built using partitions of unity). When applied to the usual smooth structure of $\mathbb{R}^n$, does it coincide with the concept ...
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1answer
18 views

$g$ is differentiable and $g'(y)=\int_{\mathbb{R}}ixf(x)e^{iyx}dm(x)$

Let $f \in \mathcal{L}(\mathbb{R},\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ measurable Lebesgue. Asumme that $x\to f(x)$ is measurable. For $y \in \mathbb{R}$ define: ...
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2answers
38 views

$\int_{0}^{\infty} x^{a-1}(e^x-1)^{-1}dx $

I need to show that I= $\int_{0}^{\infty} x^{a-1}(e^x-1)^{-1}dx = \Gamma(a) \times \Sigma n^{-a}$ where $a > 1$ I have no clue how to approach ! I am using $\Gamma(a) = \int_{0}^{\infty} ...
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15 views

Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi d\lambda$; where ...
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1answer
24 views

Lebesgue's differentiation theorem for all points

Let $f \in L^2(0,T)$ be such that $f(t)$ is well-defined for every $t$ (not just a.e. $t$). But I have no continuity of $f$. We have by Lebesgue's differentiation theorem that $$\lim_{a \to ...
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1answer
48 views

Does $\int_0^\infty f(x) dx = \lim_{n \to \infty} \int_0^n f(x) dx$ for $f \geq 0$ or $f$ not positive?

Suppose $f$ is measurable. Does $\int_0^\infty f(x) dx = \lim_{n \to \infty} \int_0^n f(x) dx$ for $f \geq 0$ or $f$ not positive? If we require $f(x) \geq 0$, the equality holds by Lebesgue's ...
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1answer
27 views

Integral on set sequence is not convergent to their countable intersection

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$ is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and denote ...
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0answers
115 views

Is Stokes theorem on $n$-cells equivalent to Stokes theorem with singularities

Could you tell me if Stokes theorem for $n$- cells is equivalent to Stokes theorem for manifolds with singularities (where the set of singularities has measure zero/ is negligible)? I mean, would it ...