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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18 views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
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0answers
14 views

Heuristic: Daniell integral vs. Lebesgue integral

What are the advantages of the Daniel Integral over the Lebesgue integral and visa-versa? Heuristically speaking, I was wondering why this axiomatic operator is less popular besides the fact that it ...
3
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1answer
115 views

How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space

Let $\mathcal{L}^2[(0,1)]$ denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1]. Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space. I believe that I can ...
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1answer
32 views

Evaluating an integral by dominated convergence theorem

I would like to know how to solve this two problems: a) $$ \lim_{n\to \infty}\int_0^n \left( 1-\frac{x}{n} \right)^{-n}\log{(2+\cos(x/n))} \, dx $$ b) $$ \lim_{n\to \infty}\int_0^{\infty} n e^{-nx} ...
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3answers
32 views

Showing that a sequence of Lebesgue Integrable functions converges to 0 a.e

Let $f_n: E \to \mathbb{R} \cup \{\infty\}$ be Leb.-integrable and suppose: 1) There is a sequence $\{a_n\}$ s.th. $a_n \ge 0$ 2) $\sum_{n=1}^{\infty} a_n = L$ (i.e.: it converges to some L) ...
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1answer
43 views

Computing Lebesgue Integral

I am trying to show that \begin{equation} \int_0^{\infty} \frac{x}{e^x-1} dx = \sum_{n=0}^{\infty} \frac{1}{n^2} \end{equation} Please note that this is the Lebesgue integral. My current strategy ...
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2answers
55 views

Show that the following function is Lebesgue integrable.

\begin{equation} \int_0^{\infty} \frac{x}{e^x-1} dx \end{equation} I know that this function has been tackled from other perspectives, but I haven't been able to find anything on its Lebesgue ...
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1answer
41 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization ...
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1answer
18 views

Why (and how) to choose $a$ in $\varrho(t):=a 1_{[0,1]}(t)\exp\left(\frac 1{t^2-1}\right)$ such that $\int_{\mathbb{R}^n}\varrho(|x|)\;dx=1$?

Let $$\varrho(t):=\begin{cases}\alpha\exp\left(\frac 1{t^2-1}\right)&\text{, if }t\in [0,1]\\ 0&\text{, otherwise}\end{cases}$$ Why (and how) can we choose $\alpha\in\mathbb{R}$ such that ...
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1answer
18 views

Calculate the Lebesgue integral of a step function

I'm having some trouble with this problem. Let $$f(x)= \begin{cases} 1 &\text{for}\,\, x = \frac{1}{n}\,,\, n=1,2,\cdots \\ 2 &\text{otherwise} \end{cases}$$ Compute the value of the ...
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0answers
19 views

Sion Minmax theorem for integral operators

Suppose $f, g\in S=L^p([0,1],\Sigma,\mu,[0,1])$. The objective $L:S\times S\to R$ is given by $$L(f,g)= \int f (h-g) d\mu, $$ where $h\in S$ is fixed. Could we apply Sion Minmax theorem to conclude ...
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3answers
49 views

Sequence of Lebesgue integrals

I am trying to solve this problem: Let $f \in L^1([0,1])$ be a non negative, finite function. Show that $$\lim_{n \to \infty} \int_0^1 \sqrt[n]{f(x)}dx=m(\{x \in [0,1]/f(x)>0\}$$ This is what I ...
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1answer
34 views

$Var(F)|_{a}^{b}=\int_{a}^{b}|f|d\alpha.$

Let $[a,b]$ be an interval in $\mathbb R$,and $\alpha :[a,b]\to \mathbb R$ be monotone increasing. Let $f:[a,b]\to \mathbb R$ be integrable, bounded and with respect to $\alpha$. Define $F:[a,b]\to ...
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1answer
47 views

Real Analysis and Lebesgue Measure: Step Functions

For each integer $n$ and $x\in (0,1)$, let $x = 0.k_1k_2k_3...k_nk_{n+1}...$, where $k_i$ is an element of $\{0,...,9\}$ be the decimal expansion of $x$. For such $x$, define $f_n(x) = ...
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1answer
32 views

Limit of sequence of Lebesgue integrals

I am trying to prove the following: Show that for each $g \in L^1([0,\infty))$,$$\lim_{n \to \infty}\dfrac{1}{n} \int_0^n xg(x)dx=0$$ What I did up to now is: Let $h(x)=xg(x)$, then $|h| \leq n|g|$ ...
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2answers
38 views

$f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable

I know there are similar questions up proving this, but I had a question specific to the following proof (specifically in bold): Let $f$ be a Lebesgue measurable function on $\mathbb{R^n}$. Then the ...
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2answers
15 views

Mean value theorem for sliding window of Lebesgue integral of integrable function

Take $f \in L^1(\mathbb{R})$ and define $g(x) = \int_x^{x+1} f(t) \, dt$. If $g(a) > 0$ and $g(b) < 0$, is it necessarily true that there is some $c \in [a,b]$ such that $g(c) = 0$? I feel as ...
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2answers
77 views

If a function is Lebesgue measureable, does this imply Lebesgue integrability?

Say we take the measure of a countable set, we obtain that $\mu=0$. Now if this is the case, does this automatically imply that it is Lebesgue integrable as well? The reason I bring up the set is ...
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1answer
23 views

Lebesgue integrable function at $[0,+\infty)$

Let $f:[0,+\infty) \to \mathbb R$ be a Lebesgue integrable function. Prove that there is a sequence $(x_n)_{n\geq 1}$ such that $\lim_{n \to \infty}x_n=+\infty$ and $\lim_{n \to \infty}f(x_n)=0$. ...
0
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1answer
24 views

Lebesgue integration by parts in Sobolev space $W^{1,2}(\mathbb{R})$

Let $\phi, \psi \in W^{1,2}(\mathbb{R}) \subset L^2(\mathbb{R})$ and we want to integrate by parts the following piece: $$\int_{\mathbb{R}}\phi(x)\psi'(x)dx$$ Supposedly, it should look like this: ...
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1answer
37 views

Characteristic functions dense in simple functions in $L^1$?

Consider $L^1(X,Y)$ where $X=Y=[0,1]$ and $ \langle f, g\rangle =\int fg $. Is the set of characteristic functions $\{\chi_{A} \}$ dense in the set of simple functions $\{ s\}$, in the sense there ...
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0answers
16 views

Lebesgue integral over subset

I have a basic question related to Lebesgue integration restricted to a subset of $\mathbb R^n$. In general, for $(X,\Sigma,\mu)$ a measurable space, if $f$ is measurable one can calculate the ...
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2answers
34 views

Application of Monotone Convergence Theorem

Suppose $f ∈ L^{1}([0, 1])$. Prove that $lim$ $ε→0^{+} \int_{[0,ε]} f dµ = 0$ My attempt at proof: Let $B_N$ be an open ball of radius $N$ centred at origin. $E_N:=$ {$x: f(x)\leq N$} ...
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0answers
32 views

Show that this integral is finite $\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx$

Let $p > -1$ and $r \in \mathbb{N}$, show that $$\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx = \int_0^\infty x^p (\ln x)^r e^{-x} dx$$ and that this integral is finite. To ...
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2answers
35 views

Isn't the statement of the Fatou's lemma somewhat problematic?

My lecture notes define $\int f := \int f^+ - \int f^-$ provided both $\int f^{\pm}$ are finite. And then the Fatou's lemma is stated in the following way: Let $f_n$ be a sequence of integrable ...
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2answers
49 views

Why is it important that $L^P$ spaces be complete?

I know that Banach spaces are ubiquitous and incredibly important in a lot of areas of math, but I was hoping for an intuitive explanation as to why (and when) it's important in the case of $L^p$ ...
2
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1answer
53 views

If $f_n$ converges in measure to $f$, prove prove $\lim_{n\to\infty} \int_a^b f_n(x)dx=\int_a^b f(x)dx$.

Let $\{f_n\}_{n=1}^\infty$ be a sequence of measurable functions that converges in measure to $f$ on $[a,b]$. If there is a Lebesgue integrable function $g$ on $[a,b]$ such that $|f_n(x)|\leq g(x)$, ...
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0answers
11 views

Judging whether a function is integrable

Here, f(x,y) is defined on [-1,1] x [-1,1]. I tried to calculate the integration of absolute value of f on the domain, using Tonelli's theorem. But the function is too complicated for me to ...
3
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2answers
61 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
3
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1answer
33 views

Find $\lim_{n \to \infty} n^{\alpha} \int_{n}^{\infty} \frac{f(x/n^2)}{x^{\alpha + 1}}(x-n)dx$

I am looking at an old exam in my measure theory and integration class. I am trying to solve a problem and am wondering if I am doing it right. Problem Let $f$ be a bounded measurable function on ...
2
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2answers
41 views

Computing limits using Monotone Convergence theorem

I am trying to compute the limits of $\lim_{n \rightarrow \infty} \int\limits_0^{\infty} \dfrac{1}{(1+\dfrac{x}{n})^n \sqrt[n]x}dx $ by using Monotone convergence theorem of integrals and switching ...
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1answer
37 views

To show that a function defined by integral is absolutely continuous

Let $$ F(x)=\int_{[0,x]\times[0,x]}f,\quad x\in[0,1] $$ Here f is a Lesbegue-integrable on the unit square $[0,1]\times[0,1]$. I need to show that $F$ is absolutely continuous and express the ...
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0answers
41 views

Integral of $\frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}$ [closed]

Does the following integral have a closed form solution? $$ \int_{0}^{\infty} \frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}{\rm d}x $$ where $\alpha$, $\beta$ and $x_0$ are ...
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1answer
29 views

Translation property in $L^1(\mathbb{R})$ space

Let $g(x)$ be a bounded measurable functions on $\mathbb{R}$, and $f(x)$ be in $L^1(\mathbb{R})$. Notation: $\int_\mathbb{R} h(x)dx=\ $the integration of measurable function $h$ over $\mathbb{R}$ I ...
0
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1answer
16 views

Application of Fubini's theorem, lebesgue integral with product measure

Show that for $f(x,y)=xy/(x^2+y^2)^2 $ for $ x,y \neq (0,0) $ and $ f(0,0)=0$ , the iterated integrals $\int_{-1}^1\int_{-1}^1fdxdy $ and $\int_{-1}^1\int_{-1}^1fdydx$ coincide but that the double ...
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2answers
40 views

Prove that $\int_0^1\int_x^1 \frac{f(y)}ydy\,dx=\int_0^1f(x)\,dx$ if $f$ is Lebesgue integrable

Suppose f is a Lebesgue integrable function on [0,1] and define a new function by $$g(x)= \int_x^1 \frac{f(y)}ydy$$ for all x in [0,1]. Prove that $$\int_0^1{g(x)} dx=\int_0^1{f(x)} dx$$ My ...
4
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1answer
52 views

Convergence of $ L^{p} $-integrals implies convergence in $ L^{p} $-norm?

Let E be a measurable set, $\{ f_n \}$ and $f$ are in $L^p(E)$ such that $f_n \to f$ pointwise a.e. If $\lim \|f_n \|_p = \| f \|_p$, is it true that $\lim \| f_n - f \|_p = 0$? I have tried ...
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0answers
29 views

Prove that $\int (\delta x)=\delta^{-d} \int f$

Let $f$ be a real-valued integrable function on $\mathbb{R}^d$. Prove that $$\int f(\delta x) = \delta^{-d} \int f.$$ I let $f(x)=\chi_E(x)=\begin{cases} 1 & \text{if }\delta x \in E \\ 0 ...
1
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1answer
72 views

Is it true that $ \int_{[1,\infty)} f_n\to \int_{[1,\infty)} f$?

Can you please help me solve this on measure theory? My TA did not go over this. He said we are not going over this but you can do this if you want. Can someone please explain to me? Thanks. Suppose ...
2
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2answers
56 views

If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$?

Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that $$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? ...
3
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1answer
36 views

Intuition behind the Riesz-Thorin Interpolation Theorem

Quoting the definition on Wikipedia, Let $(\Omega_1, \Sigma_1, \mu_1)$ and $(\Omega_2, \Sigma_2, \mu_2)$ be $\sigma$-finite measure spaces. Suppose $1 \leq p_0 \leq p_1 \leq \infty$, $1 \leq q_0 ...
0
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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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0answers
16 views

Can't evaluate triple integral over tetrahedron [duplicate]

I have seen similiar but not the same question over here, but I can not reproduce this answer to my question. Help evaluating triple integral over tetrahedron I have to calculate integral: ...
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1answer
43 views

Function is identically zero almost everywhere

Prove that if $\int_E f d\mu = 0$ for some $f \ge 0$, then $f = 0$ almost everywhere. This is Execrise 1 in Chapter 11 of baby Rudin. My attempt: $\int_E f d\mu = 0 \implies$ sup { ${\int_E s ...
1
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1answer
30 views

can a LUB be part of an open interval

I am re-reading an old textbook "Introduction to Hilbert spaces and applications" by Lokenath Debnath and Piotr Mikusinski, and there is a proof of a lemma in a chapter about the Lebesgue integral ...
1
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1answer
43 views

Proving the existence of a certain Lebesgue-measurable set.

Let $ m $ be the Lebesgue measure on $ \mathbb{R} $ and $ f: \mathbb{R} \to [0,\infty) $ a Lebesgue-integrable function. Show that there exists a Lebesgue-measurable set $ E \subseteq [0,\infty) $ ...
0
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1answer
30 views

How to show that $\dfrac{\sin(x^2+y^2)}{(x^2 + y^2)^\alpha}$ integrable on $\mathbb{R}^2$

I need to show that $$k(x,y) = \dfrac{\sin(x^2+y^2)}{(x^2 + y^2)^\alpha}$$ is integrable on $\mathbb{R}^2$ for $1<\alpha <2. $ How do I go about this? I'm pretty sure I need to use Tonelli's ...
1
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2answers
331 views

Unbounded Lebesgue integrable functions

Is $f(x)=\sqrt{-2\times \ln{x}}$, $x \in (0,1)$ (inverse of the Gaussian curve) Lesbegue integrable ? If not, is there any function that tends to $\infty$ at some point $x$ and is still integrable ? ...
9
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2answers
296 views

Lebesgue integration of simple functions

Define $f : [0,1] \to \Bbb R$ by $f(x) := 0$ if $x$ is rational, and $f(x) := d^2$ if $x$ is irrational, where $d$ is the first nonzero digit in the decimal expansion of $x$. Show that ...
2
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0answers
25 views

Enigma in applying Lebesgue dominated convergenege theorem

Let $p\in \mathbb{C} \lbrack z],~p=p\left( re^{it}\right) ,n=\deg p$ and we want to compute de limit $$ \lim_{r\rightarrow\infty}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{p^{\prime}\left( re^{it}\right) ...