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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0
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1answer
26 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 ...
0
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1answer
20 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 ...
3
votes
3answers
60 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 ...
3
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1answer
48 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 ...
1
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1answer
46 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 ...
0
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1answer
56 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) = ...
1
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1answer
42 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 $|h| \leq n|g|$ ...
1
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2answers
21 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 ...
2
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2answers
42 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 ...
2
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0answers
22 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 ...
0
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2answers
83 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 ...
0
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1answer
28 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 $\lim_{n \to \infty}f(x_n)=0$. ...
1
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0answers
25 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 ...
1
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1answer
50 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 ...
0
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1answer
28 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: ...
0
votes
2answers
49 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. $E_N:=$ {$x: f(x)\leq N$} ...
2
votes
1answer
55 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 ...
1
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2answers
48 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 ...
1
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2answers
60 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 $L^p$ ...
0
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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
votes
2answers
68 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 ...
2
votes
1answer
57 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 $|f_n(x)|\leq g(x)$, ...
3
votes
1answer
34 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 ...
1
vote
1answer
40 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 ...
2
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2answers
48 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 ...
0
votes
1answer
24 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 ...
-1
votes
1answer
31 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 ...
1
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2answers
44 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 ...
1
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0answers
34 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 ...
4
votes
1answer
53 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 ...
2
votes
2answers
58 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
votes
1answer
56 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
votes
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: ...
-1
votes
1answer
47 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 ...
0
votes
1answer
34 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
vote
1answer
45 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) $ ...
2
votes
0answers
26 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) ...
9
votes
2answers
318 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 ...
0
votes
1answer
40 views

square integrable

I have a question about Lebesgue integral. Let $(S,\Sigma,m)$ be a $\sigma$-finite measure space. Let $f$ be a $\Sigma$-measurable real-valued function. If $f$ satisfies that $\forall g \in ...
3
votes
2answers
121 views

Why does Fubini's theorem not hold/apply to this function?

Given the function $$ f(x,y) = \begin{cases} e^{y-x}, & x > y \geq 0 \\ -e^{x-y}, & 0 \leq x \leq y \end{cases}$$ I have already determined that $$ \int_0^\infty \left( \int_0^\infty f(x,y) ...
0
votes
2answers
22 views

If $\lim_{n\to\infty} \int_E f_n dx$ is finite, prove $f$ is Lebesgue integrable on $E$ and $\lim_{n\to\infty} \int_E f_n dx=\int_E fdx$.

Let $\{f_n\}_{n=1}^\infty$ be a decreasing sequence of Lebesgue integrable functions defined on a measurable set $E$. Suppose there is a function $f$ such that $f_n(x)\to f(x)$ almost everywhere on ...
1
vote
1answer
41 views

Does the integral converge I can't find counterexample

I found the following question in the book of kolomogorov fomin introductory real analysis and I don't know how to solve it. Does anyone have any ideas? Suppose $f$ is integrable on sets ...
3
votes
1answer
46 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
1
vote
2answers
79 views

Do Riemann and Lebesgue integrals always agree?

I know that on a closed bounded interval, say $[a,b]$ in $R^1$, if a function is Riemann integral, then it is Lebesgue integrable, and the values of those two integrals are the same. But, is this ...
0
votes
1answer
32 views

A Quick Question on the Monotone Property of Integrals.

Let $(\Omega,\mathcal{A},\mu)$ be a measure space with $f$, $g\in \mathcal{L}_1(\Omega,\mathcal{A},\mu)$. If for any $A\in\mathcal{A}$ we have $$\int_Afd\mu\geq\int_Agd\mu\space ,$$ show that $f\geq ...
1
vote
1answer
45 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
0
votes
1answer
30 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
5
votes
3answers
73 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
2
votes
1answer
72 views

I need to prove whether two sequences are equidistributed or not

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. A sequence $\{x_{n}\}$ in $[0,1]$ is called ...