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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50 views

sup of integrals of simple functions = inf of integrals of simple functions implies f is measurable?

Let $E \subseteq \mathbb{R}$ be measurable with $|E| < \infty$, and f a nonnegative, bounded function on E. Prove that $sup \lbrace \int_E \phi : 0 \leq \phi \leq f, \phi$ simple$ \rbrace = inf ...
2
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1answer
32 views

Measure in spectral theorem always positive?

In my functional analysis lecture we introduced the continuous functional calculus on $\sigma(T)$ if $T$ is a self-adjoint operator. Then the Riesz representation theorem gives us that ...
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0answers
27 views

Lebesgue integral of non-invertable function

Edit: This questions has been sufficiently answered I have a function which is too complicated to integrate using Riemann integration so I am trying to evaluate it with Lebesgue integration. I have ...
2
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1answer
33 views

Nash inequality : does $f\in L^1$ and $\nabla f \in L^2$ implies $f\in L^2$?

Let $f$ be any function that belongs to $L^1(\textbf{R}^d)\cap H^1(\textbf{R}^d)$ ($d$ a positive integer). Nash inequality applies in this case and gives us $$\| f\|_{L^2}\leq C \| f\|_{L^1}^r \| ...
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0answers
20 views

Express $\int_A^Bfdg$ as a sum using $\sigma_n$ and $\sum_{i=0}^ns_i$.

Question Let $\{s_n\},\{\sigma_n\}\subset\mathbb{R}$ be monotone increasing and are able to be summed. Also let $\{u_n\},\{v_n\}\subset\mathbb{R}$ and $A < u_i < v_i < u_{i+1} < B$ for ...
4
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1answer
41 views

Lebesgue measure of sets [closed]

Calculate the Lebesgue measure of following sets: $A=\{(x,y): x\in\mathbb{Q} \vee y\in\mathbb{Q}\}$ $B=\{(x,y): x-y\in\mathbb{Q}\}$ So I guess I need to calculate an integral over those sets. But ...
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1answer
38 views

Stone-Weierstrass: Literature

Short question... Does someone know some textbook, a paper or notes that treats: An algebra of functions with identity that separates is dense within a function space.
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0answers
16 views

What does it mean for the fourier transform to map $L_2$ to $H_2$?

I am in a very introductory fourier transform class. When I looked up online about fourier transforming as a linear mapping, there was this reference that the fourier transform maps $L_2$ into $H_2$ ...
1
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0answers
28 views

Is my proof correct? (Lebesgue integrable functions)

Suppose $f \in \mathcal{L}^1(\Omega,\mathcal{A}, \mu).$ Prove that for each $\epsilon > 0$ there exists a bounded $\mathcal{A}$-measurable function $g$ such that $\int_{\Omega} |f-g| d\mu < ...
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2answers
20 views

combine the integrands

If the integral looks like this: $$ \iint \{\iint f(x)f(y)\mathrm{d}x\mathrm{d}y\}g(x)g(y) \mathrm{d}x\mathrm{d}y $$ is it legitimate to 'combine' the integrals as $$ \iint f(x)f(y)g(x)g(y) ...
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1answer
19 views

Borel Measures: Integral Approximation

Problem Given a locally compact Hausdorff space. Consider a regular Borel measure in the sense: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}_+:\quad\mu(E)=\inf_{E\subseteq ...
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1answer
54 views

Understing proof for the case when riemann and lebesgue-integral are the same.

There are two things in this proof by Rudin, in Principles of mathematical I don't understand, I have marked them in red, and will explain in more detail below, can you please help? Does he really ...
3
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2answers
38 views

$\int | \nabla |u||^2 = \int |\nabla u|^2$ implies $u=e^{ic} |u|$

Assume $\int_{\mathbb{R}^N}|\nabla |u||^2 = \int_{\mathbb{R}^N}|\nabla u |^2 $. Is is true that $u=e^{iC} |u|$, where $C$ is a constant.
2
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2answers
37 views

Lebesgue integral with sum

How can we swap sum under integral; $$ \sum\limits_{n=0}^\infty\int\limits_0^{\pi/2}(1-\sqrt{\sin x})^n\cos x dx $$
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2answers
37 views

How can we evaluate this with Lebesque dominated convergence theorem?

$$\lim\limits_{n\to\infty}\int\limits_0^1\dfrac{x^2}{x^2+(1-nx)^2}dx$$ shall we add and remove$$\lim\limits_{n\to\infty}\int\limits_0^1\dfrac{x^2+(1-nx)^2-(1-nx)^2}{x^2+(1-nx)^2}dx$$ and going this ...
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1answer
26 views

Looking for example $\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f \,d\lambda$ where $\mu(A) < \infty$

I'm looking for a summable non-negative function $f: \Bbb{R} \to [0,\infty)$ and a measurable set $A$ with finite measure such that $$\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f ...
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0answers
26 views

Achieve summation by parts via integration by parts on $\int_B^Afdg$

Let $\{s_n\},\{\sigma_n\}\in\mathbb{R}$ be monotone increasing and are able to be summed. Also let $\{u_n\},\{v_n\}\in\mathbb{R}$ and $A < u_i < v_i < u_{i+1} < B$. Let $H$ denote the ...
2
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1answer
64 views

Computing $\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n}\,dx$?

I am trying to compute this integral/limit, I don't feel like I have any good insight... $$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx.$$ I have tried to make a change of ...
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1answer
103 views

Is $\sin(1/x)$ Lebesgue integrable on $(0,1]$?

Is the function $f(x)=\sin(1/x)$ Lebesgue integrable on $(0,1]$? I know that, as $f$ is continuous on the set, it is a measurable function. However, I'm stumped on how to go on. A nudge in the right ...
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0answers
19 views

if the function series is uniformly convergent is it also almost everywhere convergent?

Well,I know that if the series is normally or almost everywhere convergent,it's also convergent in measure.I wonder if this kind of connection is in other convergence types.For this particular example ...
2
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0answers
53 views

Use of Fubini's Theorem

Let $f(m,n)$ be a real valued function for all $m,n\in\mathbb{N}$. Suppose that $$ \sum_{m=1}^\infty\lvert f(m,n)\rvert\le\frac{1}{n^2} $$ for each positive integer $n$. Use Fubini's Theorem to prove ...
3
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3answers
49 views

If $\Omega$ bounded, is it true that $f \in L^p(\Omega) \Rightarrow f \in L^{p+\epsilon}(\Omega)$ for small $\epsilon$?

Let $\Omega$ be a bounded subset of $\mathbb{R^n}$, and $f \in L^p(\Omega)$. Prove or disprove that there exits $\epsilon > 0$ such that $f \in L^{p+\epsilon}(\Omega)$. It seems to me that there ...
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1answer
37 views

Proving set of density points is an open set

Let $A\subset\mathbb{R}$ measurable and denote the set of density points $$\tilde{A}:=\{x\in\mathbb{R}\mid \lim_{\epsilon\to 0}\frac{m([x-\epsilon,x+\epsilon]\cap A)}{2\epsilon}=1\}$$Porve/Dsiprove ...
1
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1answer
23 views

Finding limit of integral

LEt $f:\Omega\to[0,\infty]$ measurable s.t $0<c:=\int_{\Omega}fd\mu<\infty$ . Prove that $$\lim_{n\to\infty}\int_{\Omega}n\log\bigg(1+\big(\frac f n\big)^a\bigg)d\mu=\cases{c\quad ...
0
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1answer
43 views

Equality in Hölder's inequality

In this answer, I am confused by the following step. $$\int \hat{f} \hat{g}=\frac{1}{p} \int \hat{f}^p + \frac{1}{q} \int \hat{g}^p \iff \hat{f} \hat{g} = \frac{1}{p} \hat{f}^p + \frac{1}{q} \hat{g}^p ...
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1answer
31 views

Tensor Product: Identification

This is meant as note. Given a measure space and a Hilbert space. Then there's an identification: $$\mathcal{L}^2(\mu)\hat{\otimes}\mathcal{H}\cong\mathcal{L}^2_\mathcal{H}(\mu):\quad ...
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1answer
46 views

convergence of a series - lebesgue [closed]

Let $\xi_n$ a sequence in $[0,1]$. Prove that the series $$\sum_{n=1}^{+ \infty} \frac{1}{{n^2|x-\xi_n|}^{1/2}}$$ converge for almost all $x\in [0,1]$ (Lebesgue measure)
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1answer
93 views

Measure theory - real analysis [closed]

Suppose μ is a positive Borel measure in $[0, +\infty)$ such that μ($[0, +\infty)$)=1. Prove that $$\int_{0}^\infty (1-\mu([0,x)))dx= \int_{[0, +\infty)} x\,d\mu(x)$$ where $dx$ is the Lebesgue ...
0
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1answer
44 views

lebesgue measurability of $\sum_{n=1}^\infty \frac{\cos nx}{n^2}$ [closed]

Can u show that Lebesgue measurability of $$f(x)=\sum_{n=1}^\infty \frac{\cos nx}{n^2}$$ and $$\int\limits_0^\pi f(x)\,dx=\text{?}$$
6
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3answers
167 views

a characterization of $L^p$ space

The following question should be part of the questions I recently asked here Prove or disprove a claim related to $L^p$ space If $g \in L^p(\Omega, \lambda)$ where $\Omega$ is a bounded subset of ...
2
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1answer
90 views

An exercise on an integral inequality

I need help with this exercise: let $f,g,h : [0,1] \to [0,\infty] $ integrable functions. Prove that the following statements are equivalent: i) $(f(x))^2 \leq g(x)h(x) $ almost everywhere. ...
2
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1answer
30 views

Justification of the use of Fubini's theorem in $\int_{(0,A)} \big[\sin(x) \int_{(0,\infty)}\exp(−nx)dn\big] \,dx.$

I am trying to figure the value of $$\lim_{A \to \infty} \int_{(0,A)} \frac{\sin(x)}{x} \,dx$$ using Fubini's theorem. As for now, I am thinking to substitute $1/x = \int_{(0,\infty)} \exp(−nx) \, ...
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0answers
33 views

What is the value of Lebesgue integral $\int_{E}{(f(x) - 1) dx}$?

I am trying to solve this problem: If $f : \mathbb{R} \to \mathbb{R}$ is a Lebesgue measurable function and $\int_{0}^{1}{f(x)dx} = 1$ (Lebesgue integral) and $E = \{x \in [0, 1] \mid f(x) > 1\}$, ...
3
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1answer
46 views

Convergence of series of integrals

Let $\phi \in C^\infty(\mathbb R)$ be a function such that $\phi(x), \phi'(x) \to 0$ as $x \to \infty$. I want to show that $$\lim_{n \to \infty} \int_\mathbb R \cos(nx) \phi(x) \ dx = 0$$ Doing it ...
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0answers
21 views

Computation of a Lebesgue-Stieltjes integral

I am asked to compute the integral $\int_{(0,3a]}x\,dF(x)$ with $a > 0$ where $$F(x) = \begin{cases} \pi & 0\leq x < a\\ 4+a-x & a\leq x < 2a \\ (x-2a)^2 & ...
2
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2answers
72 views

Continuity of a Lebesgue indefinite integral over unbounded interval

We know that if $f : [a,b] \rightarrow \mathbb{R}$ is Lebesgue-integrable, then $$ F(x) = \int_{a}^{x} f(t) dt $$ is continuous. But if $f : \mathbb{R} \rightarrow \mathbb{R}$ is ...
4
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1answer
45 views

Requesting hints for showing that some function that is locally $L^p$ integrable is in $L^1(\mathbb{R})$.

Suppose $\int _a^b\vert f\vert^p<\infty$ for some $p\ge 1$ and for all $a,b\in \mathbb{R}$, and for some $a>p-1$ $$\int_{2\vert y-x\vert \le x}\vert f(y)\vert ^pdy\le \vert x\vert^{-a}$$ when ...
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1answer
40 views

Two Definitions of Lebesgue Integral

So the definition of Lebesgue integral as I understand it is as follows: Let $(X, \mathcal{F}, \mu)$ be a measure space, and $f: X \to [0, + \infty]$ a non-negative function. Then for simple ...
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1answer
47 views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
3
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0answers
37 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
1
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1answer
33 views

Complex Measures: Absolute Continuity [closed]

Note: This is a lemma for: Spectral Measures: Riemann-Lebesgue Given a positive measure: $$\lambda:\mathcal{A}\to[0,\infty]$$ Consider a complex measure: $$\mu:\mathcal{A}\to\mathbb{C}$$ How to ...
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1answer
34 views

Does the mean integral over B(x,r) of a L1 function u converge a.e. to u(x)?

Suppose $u\in L^1(\Omega )$. Let $u_{x,r}$ be the mean of $u$ over the ball $B(x,r)$ (s.t. $B(x,r) \subset \Omega$), i.e. $ u_{x,r} := \frac{1}{|B(x,r)|} \int_{B(x,r)} u(y) dy$. Is it true that ...
0
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0answers
34 views

Lebesgue integral of cartesian product of functions

Given two Lebesgue Integrable functions $f,g$, is there a notion of the integral $$\int_A f \times g \, \, dx_1 \times dx_2 ?$$ Is this even a definable notion? I couldn't find anything on the ...
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1answer
33 views

Double integral of $e^{3+y^2}$ over a triangle

Evaluate $\iint_{A}^{} e^{3+y^2}dxdy$ where $A$ is a triangle with vertices $(0,0)$, $(0,-1)$ and $(1,-1)$. I don't know how to bite that. I tried multiplying it by $e^{x^2}$ and then changing the ...
1
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1answer
44 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
3
votes
2answers
104 views

Finding integral over inconvenient set

Put $F = \{ (x,y) \in \mathbb{R}^2 : |x^2-y^2| \leq 1, 2|xy| \leq 1 \}$. How do we find the following integral? $$\int_F (x^2 + y^2) \,d(x,y)$$ I'm sure we need to use Jacobi's transformation ...
1
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0answers
33 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
1
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1answer
31 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
3
votes
1answer
52 views

Computing with Lebesgue integrals

This problem comes from Royden's Real Analysis, 4th ed., pg 84, #19: For a number $\alpha$, define $f(x)=x^\alpha$ for $0<x\le 1$ and $f(0)=0$. Compute $\int_0^1 f$. MY WORK: I know ...
2
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1answer
19 views

Sequence of continuous functions convergent to $0$ with the integral equal to $1$

I am looking for a sequence of continuous functions $\{f_m\}$ defined in $A\subset\mathbb{R}$ with $\lim\limits_{m\to\infty} f_m=0$ such that $\int_A f_m \;d\mu=1$. The problem I have is with the ...