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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0answers
12 views

Integral of convex function applied on a function

Let $f$ be an integrable function of $\mathcal{L}(\mathbb{C},\mathbb{R})$, measure Lebesgue. I want to prove that there exists an increasing convex function $H:\mathbb{R}^+\rightarrow\mathbb{R}^+$ ...
0
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1answer
32 views

Can someone solve my non-understandable process in proving a theorem?

Theorem. Let $E$ be a subset of $\mathbb{R}^n$. Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$. For your ...
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0answers
15 views

Interesting measure theory property in L^p

Let $f, f_n \in L^p (X)$, so that there is a function $g\in L^p (X)$ with $|f_n|\leq g,\ \forall n$ and $\forall \epsilon>0, \lim_{n\to\infty} \mu (\{x\in X\big | |f_n (x)-f(x)|\geq \epsilon\})=0$. ...
0
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1answer
26 views

Lebesgue integral, path connected and compact function

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
1
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1answer
13 views

$A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$

Let A be a real set then is it true that $A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$.
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1answer
34 views

How to evaluate the Lebesgue integral of the Heaviside function?

I have to evaluate the Lebesgue integral $$ I = \int\limits_{[-1, 1]} \chi(x) \chi(x - \frac{1}{2}) d\left(\chi(x)\chi(x + \frac{1}{2})\right) $$ where $ \chi $ is the Heaviside function: $$ ...
2
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1answer
11 views

If $f_n$ is Lebesgue integrable and $f_{n}$ converges pointwise to $f$ then is $f$ Lebesgue integrable?

If $f_n$ is Lebesgue integrable and $f_{n}$ converges pointwise to $f$ then is $f$ Lebesgue integrable? I know that this is false unless $f_{n}$ converges uniformly to $f$, but is there an example ...
0
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1answer
18 views

Continuity of Integration (Lebesgue)

On the theorem regarding continuity of integration: Let $f$ be integrable over $E$. If $\{E_{n}\}^{\infty}_{n=1}$ is an ascending countable collection of measurable subsets on $E$, then ...
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0answers
28 views

Relation between uniform distribution and dense curves

I have a question on the relation between space-filling curves and the joint distribution of two independent uniform random variables. Consider the probability space $(\Omega, \mathcal{F}, ...
1
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1answer
51 views

Comparison test and DCT

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
0
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1answer
58 views

$f$ integrable iff $\sum_{n=1}^{\infty} f(n)$ converges absolutely

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
5
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1answer
60 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
1
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1answer
56 views

$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$

How can I prove $$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$$ given a measure space $(\Omega,\mathfrak A, \mu)$, a non-decreasing sequence $(f_n)$ of measurable functions on ...
4
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1answer
35 views

Definition of outer Measure

As I understand it, the outer measure $\mu^{*}(A)$ is used to find the length of the smallest cover that covers $A$. However, in another definition, the outer measure is defined as the largest lower ...
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1answer
63 views

Limit of a sequence of Lebesgue integrals

Let $ f\in L^{1}(E) $ and $ \{E_n\}$ be a sequence of measurable subsets of $E$. If $$ \lim_{n\to +\infty} m(E_n) = 0$$ prove that $$ \lim_{n\to +\infty} \int_{E_n} f = 0.$$ I tried to interchange ...
3
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1answer
23 views

Lebesgue integration and uniform convergence

Let $\Omega$ be a bounded and measurable set in $\mathbb{R}$. If $\{f_n\}$ is a sequence of bounded and Lebesgue integrable functions on $\Omega$. If $f_n$ uniformly converges to $f$, then how to ...
0
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1answer
34 views

Using the monotone convergence theorem to show a function is integrable

Apply the monotone convergence theorem and the fundamental theorem of calculus to show that $f(x) = \left\{ \begin{array}{ll} x^{-a} & \mbox{if } 0 < x \leq 1 \\ \infty & \mbox{if } ...
3
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0answers
47 views
+50

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
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3answers
1k views

How to decide whether Lebesgue integral or Riemann integral?

Very often I feel very uncomfortable in dealing with integrals, since I am wondering whether the given integral is meant as a (improper) Riemann integral or Lebegue integral? For instance, the Gamma ...
1
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1answer
20 views

Non-regular measure can be represented by a regular measure

Let $X$ be a locally compact and Hausdorff space, and let $\mu$ be a positive measure on the Borel sets of $X$ (here $\mu$ is not necessarily regular). Then the linear map $L : C_c(X) \to \Bbb C$ ...
3
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1answer
12 views

Product of weak and strong convergent sequences in $L^p$

I already saw some proofs here with $b_n\to b$ in $L^2$ and $a_n\rightharpoonup a$ in $L^2$. Then $$ \int a_n b_n \to \int a b. $$ But what goes wrong if both sequences are weak convergent? Proof: ...
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0answers
16 views

How is justified the derivation under the integral sign $\frac{d}{d\sigma} \left( \Re\frac{1}{\zeta(s)} \right) $?

Taking $\sigma=\Re s>1$ (this is we take $s=\sigma+it$, $\sigma$ and $t$ real numbers) then the using theknown integral representation for $\frac{1}{\zeta(s)}$, where $\zeta(s)$ is the Riemann ...
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0answers
17 views

Integrals as Signed Measures (and vice Versa)?

1. Can every integral (with respect to an integrable function) be written as a signed measure? And does the function’s decomposition into positive and negative parts align somehow with the ...
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0answers
20 views

Principal value integral of complex exponential

I'm reading the article Brownian distance covariance and stumbled upon a equality I can't seem to derive myself. We are first presented with the following lemma: and after stating this lemma, the ...
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0answers
19 views

Measurability and integrability of set and function

My textbook said: Let $E\subset\mathbb{R}^n$, let $G$ be an open set, and let $|\cdot|_e$ denote outer measure. if $\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given ...
0
votes
1answer
25 views

$f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. Find $\lim \int f_n d\lambda$.

Let $X = \mathbb R$, $\textbf{X} = \textbf{B}$ and $\lambda$ the Lebesgue measure on $\textbf{X}$. I have the following: $f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. I need to find the following: ...
1
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3answers
44 views

$f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$

If $f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$. I know that if $E$ is of finite measure, then we can infer from $f(x)\in L^2(E)$ to get $f(x)\in L^1(E)$. However, now ...
-1
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0answers
34 views

Prove that h(x) is measurable and calculate the integral of h [closed]

Let $A,B\subset \mathbb {R}$ Lebesgue-measurable sets. I want to prove that the function $h(x)=|(A-x)\cap B)|$ is measurable and that $\int_{\mathbb {R}}h(x)dx=|A|.|B|$, where $|•|$ is the ...
1
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1answer
50 views

$T$ is an $L^2$-bounded operator; find its norm

We have the integral operator $$ P:L^2(\Bbb R^n)\to\{\text{meas.functions}\;:\;\Bbb R^n\to\Bbb R\} $$ defined as $$ Tf(x):=\int_{\Bbb R^n}L(x,y)f(y)\,dy $$ where $L$ is a measurable function on $\Bbb ...
2
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1answer
37 views

Let $f$ be positive and Lebesgue measurable on $[0,1]$. Show that $\inf_{\lambda(E)\geq \epsilon} \int_E fd\lambda >0$ for any $\epsilon\in(0,1]$.

The title says it all. I've already shown, for an earlier part of this problem, that for any $E$ with $\lambda(E)>0$, we have $\int_E fd\lambda >0$. I did that by reductio, showing that ...
-4
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0answers
17 views

Property about Lebesgue Integral for Bounded measurable function [closed]

[![property][1]][1] f and g bounded measurable functions defined on a set E of finite measure. How can I prove that property? Characteristic function of E1 union E2 is the sum of characteristic ...
8
votes
2answers
75 views

Show that the set $\{x \in \mathbb{R}| \lim_{n \to \infty} \sin(a_n x) \mbox{ exists}\}$ has zero measure

$a_n$ is a sequence of real numbers such that $a_n \to +\infty$. Show that the set $E = \{x \in \mathbb{R}| \lim_{n \to \infty} \sin(a_n x) \mbox{ exists}\}$ has zero (Lebesgue) measure. The hint for ...
0
votes
1answer
26 views

Lebesgue Integral-Question

Hi guys, How can I evaluate Lebesgue Integral of this function. I think first I should show that is simple function ?
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4answers
70 views

$\int_\Omega f d\mu = 0 $ if and only if $f(x)=0$ almost everywhere

can someone give me a hint on what kind of theorem/definition I should make use of to solve this? Let $(\Omega,\mathfrak A, \mu)$ be a measure space and $f:\Omega \to \mathbb R$ a non-negative ...
0
votes
2answers
35 views

Where $f:[0,1]\to\mathbb R$ is Lebesgue integrable, show $\lim_{n\to\infty} n\lambda(\{x:|f(x)|\geq n\})=0$.

Title says it all. It's clear why $\lim_{n\to\infty}\lambda(\{x:|f(x)|\geq n\})=0$ -- since otherwise for arbitrarily high $n$ there'd be a subset of $[0,1]$ with nonzero measure where $|f|\geq n$ and ...
0
votes
0answers
8 views

What's the function that it is neccesary to show being bounded locally integrable in the Wiener-Ikehara Theorem?

When I am reading in (this video from an official channel in You$\color{red}{\text{Tube}})$ mathscienciechannel, that has the most high quality, in my attempt to understand the facts that currently I ...
1
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1answer
27 views

Convergence of sequence of smooth functions

I have the following $\{f_n\}^\infty$ sequence of smooth functions where $f_n:[0,1] \to \Re$ and $f_n(0) = 0$ with the following assumptions: $$ f_n(x) \to f(x)\ \forall x \in [0,1] $$ $$ f_n' \to g ...
0
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0answers
14 views

Divergence test for a double integral $\int \int |f| dxdy$

lets say $\int (\int f) dxdy \ne \int (\int f) dydx $ can we conclude $\int \int |f| dxdy$ diverge? $f$ is assumed to be measurable over $x,y$ and $(x,y)$.
0
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1answer
29 views

Lebesgue Integral over vanishing interval

Let $f(x)$ be a Lebesgue integrable function. Then is it true that $$ \lim_{\epsilon\to 0}\int_0^\epsilon f(x)\,dx=0 $$ always? When $f(x)$ is bounded answer is trivial, but if we wish to show this ...
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votes
2answers
63 views

I need an explation of what the question wants and a ful how to answer [closed]

Consider the integral expression in x x $$P=x^3 + x^2 + ax + 1$$ where a is a rational number, At $a=?$ the value of P P is a rational number for any x x which satisfies the equation $x^2 + 2x ...
2
votes
2answers
33 views

Does $f_n \rightarrow f$ $\mu$-a.e. and $\lim \int f_n \rightarrow \int f$ imply $f_n \rightarrow f$ in $L^1(\mu)$?

The full question from this practice qual: True/False: Let $(X,M,\mu)$ be any measure space. If $f_n,f \in L^1(\mu)$ are measurable functions, $f_n \rightarrow f$ $\mu$-a.e. and $\lim \int f_n ...
1
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1answer
29 views

Is this intuitive true that $\int_Ef=\int_{I_n}f$

When proving general integral, we usually consider simple function first. For example: Simple function -->Bounded function-->Non-negative function-->General function For Lebesgue integral, in ...
1
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0answers
40 views

Regularity of a measure $n(E) = \int_{E} f(x) dx$

I would like to show that the (positive) measure $n$ on $\Bbb R \setminus \{0\}$ defined by $n(E) = \displaystyle \int_E \frac{dx}{|x|}$ is outer regular ($dx$ being the usual Lebesgue measure). ...
1
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0answers
15 views

Is the charateristic function $\chi _{\Omega }$ in the Sobolev space $W^{1,2}_{0}(\Omega)$?

Given $\Omega$ is a bounded, $C^1$ domain in $\mathbb{R}^n$. $\chi _{\Omega }(x)$ is the characteristic function of $\Omega$. I have done the followings: We can get $\chi _{\Omega }(x) \in ...
1
vote
1answer
15 views

translation invariant of integral on $\mathbb{R}$

Problem 1: Let $f\in L^1(\mathbb{R})$, show $\int_\mathbb{R}f(t)dt=\int_{\mathbb{R}}f(x+t)dt,\forall x\in (-\infty, \infty)$. Problem 2: Let $f\in L^1(\mathbb{R})$,show $\displaystyle ...
2
votes
0answers
47 views

Understanding principal value integral

I'm reading the original article on distance covariance (link), and throughout the article the author uses the following lemma: Can someone please explain what he actually means by "principal value ...
1
vote
2answers
62 views

If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.

I'm learning about measure theory, specifically the Lebesgue intregal of nonnegative functions, and need help with the following problem. Let $f:\mathbb{R}\to[0,\infty)$ be measurable and $f\in ...
2
votes
2answers
60 views

Calculate limit with integral

Hi I have a problem with following limit: $$\lim_{x\rightarrow\infty}e^{-x}\int_{0}^{x}\int_{0}^{x}\frac{e^u-e^v} {u-v}\ \mathrm du\ \mathrm dv$$ as a hint i got that i should use de l'Hospital. So: ...
2
votes
1answer
26 views

Why is the fact true that if $E$ has measure zero, then $E_a$ has measure zero?

I am proving the following theorem: For a subset $E$ of $\mathbb{R}^n$, if $E$ is measurable, then $E_a$ is measurable and $|E_a| = a|E|$. where $E_a := \{(\mathbf{x}, y):\mathbf{x}\in E, 0\le ...
-1
votes
1answer
36 views

Interesting relationship between cardinality and Lebesgue outer measure

If two sets $A$ and $B$ defined on bounded intervals have the same cardinality and $ A \bigcap B $ is non empty and the Lebesgue outer measure of A is greater than zero. Is it then true that the ...