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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44 views

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Let $f$ and $g$ be integrable functions on a measure space $(X,Σ,μ).$ For each $t ∈ \mathbb{R},$ consider the sets $F_t =\{x∈X :f(x)>t\}, G_t =\{x∈X :g(x)>t\}.$ Prove that $\int|f − g| = ...
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0answers
23 views

Change of variable for Lebesegue Integral

Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds ...
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3answers
28 views

Square Integrable and Continuous

I have come across the notation $L^2(\Omega) \cap C(\Omega)$; while I believe understand the resulting behavior, I can't get my head around the machinery. When I think about $f \in C(\Omega)$ I am ...
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1answer
59 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
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1answer
74 views

Suppose that all the functions ${f_n},f$ are integrable. Is $lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Let ${f_n(x)}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f(x).$ Suppose that all the functions ${f_n},f$ are integrable. Is ...
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117 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
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36 views

Question about a theorem concerning the continuity of integral functions

If we have $$F(t):=\int_V f(t,x)dx$$ where $V$ is some measurable subset of $\mathbb R$ and $x\mapsto f(t,x)$ is a measurable function. Moreover let $F$ be defined for all $t\in U$ a open subset of ...
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0answers
25 views

Question on Integrability of a function.

Let $(f_j)_{j\in\mathbb{N}}$ be a sequence of real functions in $L^p(X,\mathcal{A},\mu)$, where $p\geq1$. If we know that ...
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2answers
72 views

Support of a positive measure

Let $\mu$ be Lebesgue measure on $\mathbb{R}$ , $\mathcal{M}$ be Lebesgue $\sigma-$ algebra and $f\in C_{c}\left(\mathbb{R}\right)$ (continuous with compact support). Suppose $f\geq0$ over ...
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1answer
61 views

Special case of Fatou's lemma

Hi everyone: Suppose that $D$ is a domain of $\mathbb{R}^{n}$ $(n \geq1)$, $Y$ a locally compact topological space, and $\mu$ a measure on $Y$. Let $f(x,y): D\times Y\rightarrow[0,+\infty)$ be ...
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3answers
70 views

Showing that $\frac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is lebesgue integrable

I'm having some real trouble with lebesgue integration this evening and help is very much appreciated. I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over ...
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1answer
47 views

Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$

I need some help on the following real analysis past qual problem. I would appreciate some help. Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$ ...
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1answer
242 views

$\sum_{n = 1}^\infty f_n =f.$ Prove that $f′_n =f′$ a.e.

I am studying for a real analysis qualifying exam. Was hoping that there was a very slick proof for this? Thanks. Let $f_1, f_2, . . . , f : [0, 1] → \mathbb{R}$ be non-decreasing right-continuous ...
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1answer
34 views

Showing function is lebesgue integrable

I have a function $f(x) = \frac{\sin(\frac{1}{x})}{1+(\log(x))^2}$ and I am trying to find whether this is Lebesgue integrable on $[1,\infty)$. I'm really not sure where to start on this one. It ...
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3answers
197 views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
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0answers
42 views

Functions which can not be integrated via Riemann.

I am looking for some (possibly exotic) functions which can not be integrated via Riemann integration but can be integrated via Lebesgue. I am aware of the rational indicator function, i.e., $$ f(x) = ...
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2answers
26 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
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1answer
47 views

Looking for “explicit” integrals solvable using lebesgue integration theory

I am preparing for an exam in Measure and Integration Theory (Lebesgue Integration). As far as I know my professor prefers to ask students solving explicit integrals which can be solved using the main ...
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1answer
44 views

Reversing limits in Lebesgue integration

I know that reversing limits of Riemann integration is possible by putting minus sign. My question is that there is a similar result for Lebesgue integral as well. For example, $$ ...
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0answers
43 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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2answers
91 views

If $f$ and $f'$ are integrable, then $f'$ has integral $0$

I have the problem Prove that if $f$ and $f'$ are Lebesgue integrable over $\mathbb{R}$, then $\int_\mathbb{R}f' = 0$, where $f'$ is defined everywhere. Honestly, not sure where to start. I had ...
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1answer
68 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
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1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
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1answer
34 views

Monotone property of Lebesgue Intergral

Let $(X,\mathcal{A},\mu)$ be a a measure space, and $f$ and $g$ two measurable functions. Now if $f$ and $g$ are nonnegative and $f\leq g$, it can be easily seen that $\int f\,d\mu\leq \int g\,d\mu$, ...
2
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1answer
53 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
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69 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
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1answer
38 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
3
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1answer
91 views

Properties that hold almost everywhere

Let $(X,\mathcal{A},\mu)$ be a measure space and $u$ some measurable function. If $v$ is a function and $u=v$ a.e.; when is true to say that $v$ is measurable? Also let P be some property, and $f$ ...
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1answer
49 views

Condition for integrability on finite measure space

Let $(X,\mathcal{F},\mu)$ be a finite measure space. If $f:X\rightarrow \mathbb{R}$ is a measurable real function, show that, $f\in L^1(\mu)$ iff $\sum\limits_{n=0}^{\infty}\mu(\{f\geq ...
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1answer
17 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
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1answer
56 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
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2answers
82 views

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too?

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too? My outline is the following: Since $X,Y$ is a measure zero set, ...
2
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1answer
55 views

Integral equation $f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy$

I'm trying to solve the following equation $$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0 $$ where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on ...
2
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1answer
58 views

Calculating a Lebesgue integral

Calculate the Lebesgue integral of, $$\lim_{n\to\infty}\int_{[0,1]}\frac{n\sqrt{x}}{1+n^2x^2}$$ I know I should use the Lebesgue dominated convergence theorem but what should be the dominating ...
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1answer
64 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
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2answers
463 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
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0answers
32 views

Tensor Product: $\mathcal{L}^2(\Omega_1)\hat{\otimes}\mathcal{L}^2(\Omega_2)\cong\mathcal{L}^2(\Omega_1\times\Omega_2)$

I have given the realization: $$(f\otimes g)(x,y):=f(x)g(y)$$ I want to prove that the tensor product of Lebesgue spaces is isomorphic to the product Lebesgue space: ...
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2answers
41 views

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as ...
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1answer
71 views

Inequality for integral => Inequality for integrand

I have that for any measurable set $\Omega\subset\mathbb{R}^d$ with $|\Omega|<\infty$ \begin{align}\sqrt{\int_\Omega f(x) dx }\leq \sqrt{c\cdot|\Omega|} + \sqrt{\int_\Omega g(x) dx }.\end{align} ...
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1answer
38 views

$C_c(\mathbb{R})$ is dense in $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$… right?

The intersection $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ is (allegedly) a Banach space for the norm $\|f\| = \|f\|_1 + \|f\|_2$. Is it also true that $C_c(\mathbb{R})$ is dense with respect to this ...
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1answer
33 views

Approximate an $L^2$ function from “inside”

Consider a bounded domain $\Omega \subset \mathbb R^d$ and a function $f \in L^2(\Omega)$. Now $f$ can be approximated through a sequence of functions $f_n \in H^1(\Omega)$ (or even ...
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1answer
49 views

Dini Derivative

Let $f$ be defined on $\mathbb{R}$ such that $$ f(x) = \begin{cases} |x|, & \text{if }x \in \mathbb{Q} \\ |2x|, & \text{if }x \notin \mathbb{Q} \end{cases} $$ Calculate ...
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2answers
81 views

Integral $=0$ implies function$=0$

Let $f:[a,b]\to\mathbb{R}$ be a mesurable function. How can we show that if $$\int_a ^xf(s)ds=0,$$ for all $x\in[a,b]$, then $f=0$.
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0answers
21 views

Jordan content under continuous differentiable map

I have the following problem which seems simple but in fact I find no proof for it so I am wondering if I could get some help. Let $A$ be a compact set subset of an open set $U$ in $\mathbb{R^n}$, ...
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27 views

Bounded $L^1$ functions subgroup of $L^2$

I'm currently looking at statistics and the characteristic function. And the claim is that the characteristic function must exist for every probability distribution since every probability ...
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0answers
49 views

Proving that $f_n(x)=x^n$ does not converge on $[0,1]$ under the $L_1$ metric

We see that the pointwise limit is $f(x)=1$ if $x=1$, and $f(x)=0$, for $x\in[0,1)$. (our background space is $C[0,1]$) How would one go about proving if $f_n\to f$? My feeling is that we do not have ...
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0answers
48 views

An “academic” question on integral operators

This question is motivated by another one, asked by Cameron Williams: Adjoint of an integral operator Let us say that a Borel function $k:\mathbb R\times \mathbb R\to\mathbb C$ defines an operator on ...
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0answers
46 views

Series of functions as Lebesgue integral with counting measure

Let $f_n:\mathbb{R}\to\mathbb{R}$ for $n=1,2,...$. Then \begin{equation}\sum_{n=1}^\infty f_n = \int_{\mathbb{N}}f_n\text{ d}\mu \end{equation} where $\mu$ is counting measure on $\mathbb{N}$. I ...
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1answer
53 views

Lebesgue Integral calculating problem

$$ \lim _{ n->\infty }{ \int _{ 0 }^{ 1 }{ { (1+nx^{ 2 })(1+x^{ 2 })^{ -n }\quad }dx } } $$ Please help me calculating the limit. Integral is Lebesgue Integral and what I learnt is Bounded ...
3
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3answers
172 views

Are all measures Lebesgue-Stieltjes measures?

In our lecture we ran out of time, so our prof told us a few properties about measure: He said that a measure is $\sigma$-additive iff it has a right-side continuous function that it creates. And he ...