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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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
30 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
63 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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1answer
160 views

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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137 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
40 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
42 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
43 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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3answers
108 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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40 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
88 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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32 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
38 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
28 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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21 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
39 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
60 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
45 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
34 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
51 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
58 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
28 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 ...
3
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2answers
48 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
40 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. ...
3
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1answer
49 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
64 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
65 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
31 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
46 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
39 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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0answers
27 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 $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
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1answer
27 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
50 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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1answer
85 views

Proof that for the Lebesgue indefinite integral, $\int_E fd\mu(x)=0$ implies $f=0$ almost everywhere

Can someone provide a hint for the proof of the fact that for the Lebesgue indefinite integral, $\int_E fd\mu(x)=0$ for all $E\in S$ where $S$ is a the $\sigma$ ring, implies $f=0$ almost everywhere? ...
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1answer
41 views

Inequality for convolution $| f *g| \le \|f\|_1 \|g\|_1$ [closed]

Let $f \in L^1$ and $g \in L^1 \cap C^{\infty}_c, $ is it then true that a.e. we have $$ |f *g| \le \|f\|_1 \|g\|_1?$$
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59 views

Convergence of functions in $L^1$ implies convergence of derivatives $ a.e. $?

Update: Someone gives me a good counterexample, which basically answers all the questions I posed. The example is, $$f_n(x)=\frac{sin(nx)}{n}$$ ----------------------------------------- I just came ...
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37 views

a function that is not measurable but its square is measurable

Question: Give an example of a function $f$ on $X$ to $R$ which is not $X$ measurable, but is such that the functions $|f|$ and $f^2$ are measurable. To me, $|f|$ is measurable and $f^2$ is ...
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1answer
44 views

Problem about limit of Lebesgue integral over a measurable set

This is actually problem 4T of Bartle's book "The elements of integration and Lebesgue measure". Let $f_n$, $f$ be nonnegative measurable functions on $\mathbb{R}$ such that $f_n\to\ f$ for every ...
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3answers
54 views

If $f \in L^{1}(X,\mu,\mathfrak{M},\mathbb{R})$ , $\int_{E_n}f \to \int_{X}f $ [closed]

Let $f \in L^{1}(X,\mu,\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ is the $\sigma$-algebra of Lebesgue and $\mu$ measurable Lebesgue. If $E_1\subseteq E_2\subseteq E_3 \subseteq E_4\subseteq ...
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1answer
30 views

Exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$.

For all $f:[0, 1]\to \mathbb{R}$ exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$. Any ideas. Thanks
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If $|\int f_n-\int f |\to 0 \Rightarrow\int |f_n - f| \to 0$.? [closed]

If $|\int f_n -\int f| \to 0 \Leftrightarrow |\int (f_n-f) |\to 0$ where $f_n$ and $f$ have the hypothesis of dominated convergence theorem It is true that: If $|\int f_n-\int f |\to 0 ...
2
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1answer
36 views

Proof of a double series equality

Let $b_{n_{i}} \in \mathbb{C}$ for $n,i\in \mathbb{N}$. Suppose that $$\sum_{n=1}^{\infty}\sum_{i=1}^{\infty}|b_{n_i}|<\infty,$$ then ...
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0answers
25 views

Is this Integral transformation correct?

I have an Integral: $$ \int_{-\infty}^{-y_1} \Phi(y_2)d\Phi(x_1) $$ Here: $\Phi(y_2)$ is the Gaussian density function of variable '$y_2$' which has to be integrated w.r.t Gaussian density of ...
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1answer
21 views

$f$ be Lebesgue integrable and $F = m\{f > \alpha\}$, then $F$ is right continuous

The following is a part of a problem 18.2 from from Real Analysis, N. L. Carothers: Let $f: \mathbb{R} \rightarrow [0, \infty]$ be integrable and define $F: [0, \infty) \rightarrow [0,\infty]$ by ...
2
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1answer
33 views

Showing $\left(\sum_1^\infty 2^{-n} \mathbf{1}_{x\in(0,1)}(x-r_n)^{-1/2}\right)^2$ is not integrable

A small part to a homework problem. Let $\{r_n\}_1^\infty$ be an enumeration of the rationals, and let $f(x) = x^{-1/2}$ if $0<x<1$ and $f(x) = 0$ otherwise. Define $g(x) = \sum_{n=1}^\infty ...
1
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0answers
30 views

Hardy inequality punctured space

given the minimization problem: $inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$ infimum taken on all smooth functions with compact support in the ...
0
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0answers
21 views

Differentiating the expectated area under stochastic process

I am trying to prove the following, where $X_t$ is an almost surely bounded progressivley measurable process: $$ \lim_{t\rightarrow 0}\frac 1 t \mathbb E \int_0^t X_sds = \lim_{t\rightarrow 0} ...
4
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2answers
22 views

Measurable functions such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$.

I need to find an example of a suquence of measurable functions $ f_n \geq0$ for $ n = 1,2, ... $ such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$. As I can ...