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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1answer
49 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
59 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, ...
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1answer
49 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
52 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
39 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 ...
14
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2answers
333 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
30 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
40 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
67 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
33 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
31 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
41 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
77 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
17 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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0answers
25 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
47 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
46 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
38 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
48 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 ...
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3answers
161 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 ...
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2answers
49 views

Lebesgue Integrable functions and calculating the limit

$$ \lim_{n\rightarrow \infty} \int_{\frac 1 n }^1 \frac { 1+nx }{ (1+x)^n } \, dx $$ How can I solve this problem using Bounded convergence theorem?
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0answers
25 views

a.e. convergence of dilations of a function in L^p

Let $f\in L^p(R^d)$ ($p<\infty$) and $\delta_h f(x):=h^{d/p}f(hx)$ (the normalization is so that $\|\delta_h f\|_{p} =\|f\|_p$). Consider $(h_n)$ a sequence of positive numbers such that either ...
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1answer
50 views

Lebesgue differentiation theorem with two variables

We know that if $f:\mathbb{R}\to\mathbb{R}$ is continuous then $$\lim_{h\to 0}\frac{1}{h}\int_x^{x+h} f(s)ds=f(x).$$ But if we have $f:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$, what kind of ...
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1answer
72 views

How to show that this function is differentiable?

Let $$\phi: \mathbb{R} \rightarrow \mathbb{\mathbb{C}}, s \mapsto \int_2^{\infty} \frac{e^{isx}}{x^2\ln(x)}dx$$, I want to show that this function is differentiable everywhere. Unfortunately, it ...
2
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1answer
40 views

Integral result, looking for proof or reference

I am looking for a proof or reference to a text which has the following result: Let $\Omega$ be an open subset of $\mathbb{R}^{n}$. Then if $f$ is a measurable function of some sort then if it ...
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1answer
40 views

How to prove a function is in $L_2(G)$

For $G$ is open bounded set in $\mathbf{R}^n$, $L_2(G)$ is defined as $$L_2(G) = \{f:G\rightarrow \mathbf{R} \text{ measurable}: \int |f|^2<\infty\}$$ Now, I want to prove that $f:\mathbf{R} ...
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1answer
49 views

The limit $\lim_{h\to0}\frac{1}{h}\int_0^hf(s) \, ds$

We know that if a function $f:\mathbb{R}\to\mathbb{R}$ is continuous then we have $$\lim_{h\to0}\frac{1}{h}\int_0^hf(s) \, ds=f(0).$$ What can we say if $f$ is continuous almost everywhere or ...
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0answers
25 views

Solve a problem of convergence of integral

We have F $\in$ $C(\mathbb{R}^N;\mathbb{R})$, $F\ge0$ and we have that $\int_{\mathbb{R}^N}Fdx<+\infty$. How can i prove the existence of a sequence $r_k\to+\infty$ such that $r_k\int_{\partial ...
5
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1answer
63 views

Prove $\lim_{n \rightarrow \infty}$ $\frac{1}{n}$$\int_{0}^{\space n}xg(x)dx=0$

If $g$ is a Lebesgue integrable function in $E=\lbrack 0,\infty)$, prove that $$\lim_{n \rightarrow \infty}\frac{1}{n}\int_{0}^{\space n}xg(x)dx=0.$$ I want to use the absolute continuity of the ...
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1answer
55 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
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1answer
68 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
2
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1answer
56 views

Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
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0answers
68 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
2
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0answers
90 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
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2answers
459 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
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2answers
58 views

Limit of a certain Lebesgue integral

Can someone help me to show that $\lim_{t \to \infty} \int_{\Bbb R} f(x)\sin(xt)dx =0$ for any Lebesgue integrable function $f$? Side note: How do you make the $t \to \infty$ appear directly under ...
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0answers
31 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
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1answer
82 views

Riemann-Lebesgue lemma

How can I prove the following result? Let $([-1,1],M,m)$ a measure space, where $m$ is the Lebesgue measure in $[-1,1]$. If $f$ is Lebesgue integrable, then ...
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1answer
32 views

Functions in $L^p$ and $L^q$ spaces

For any two different numbers $p,q\in[1,\infty)$ find functions $f\in L^p \setminus L^q$ and $g\in L^q \setminus L^p$. Solution: let $$f(x)=x^{-1/p}(1+|\log x|)^{-2/p}$$ Then $$\int|f|^p = ...
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1answer
68 views

Showing $\frac{\log(x)}{1+x^2}$ is Lebesgue integrable

I have calculated the integral: $\int_0^\infty \frac{\log(x)}{1+x^2} dx$ using a contour integral. However I was wondering how we would show that this is Lebesgue integrable. I have thought about ...
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1answer
46 views

Integrate over different measures

In Probability theory the expected value of a random variables $X : \Omega \rightarrow \mathbb{R}$ is defined as $E(X) = \int_\Omega X dP$ Now, if $\Omega \subset \mathbb{R}$ and has a density ...
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2answers
61 views

Why is $L^p$ isomorphic to $(L^p)^2$

Is it possible to say why the spaces in the title are isomorphic as Banach spaces? Is their a Theorem that says this or is it even possible to find an explicit representation of this isomorphism?
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37 views

Not measurable function whose module is measurable

I read through my notes that is trivial to find a not measurable function $f$ whose module $|f|$ is measurable. However I don't know how to provide such an example.
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1answer
89 views

Exercise on measure theory

Let $X\neq \emptyset$ and $f:X \rightarrow [0, \infty]$ not identical infinity. Set $$ \sum_{x \in X} f(x)= \sup \left\{ \sum_{x \in F}f(x), F \subseteq X, F \mbox{ finite} \right\}.$$ $(i)$ Show ...
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1answer
47 views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and ...
2
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1answer
37 views

Convex , then also Measurable

I was reading about Jensen's inequality and noticed that don't require $\phi$ to be measurable here: Wikipedia link. Therefore, I guess that being convex implies being measurable somehow, but I have ...
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2answers
36 views

Measurability of f(x) to g(x)

Let $(\Omega, S)$ be a measurable space. If $f:\Omega \rightarrow \mathbb{R}$ is a strictly positive measurable function and $g:\Omega \rightarrow \mathbb{R}$ is measurable show that $f^g$ is ...
2
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1answer
47 views

Computing limit with Dominated Convergence Theorem

I am trying to compute the following limit: $$ \lim_{n\to \infty} \int_0^\infty \frac{x^{n-2}}{1+x^n} \cos(\pi n x) \,dx . $$ This is a problem in an old analysis qualifying exam. Let $f_n(x) = ...
0
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1answer
48 views

Continuity of a function defined by means of the Lebesgue measure

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function and $\phi(x)=\lambda ( \lbrace{ t: f(t) >x \rbrace} )$. Prove that $\phi$ is right-continuous but not necessarily ...
1
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1answer
29 views

Reason for product sigma algebra notation

I was wondering why it is so common to denote the product sigma algebra with the same symbol that is used for tensor products. Is there a specific reason that this product symbol is used or was is ...