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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2
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1answer
88 views

An exercise on an integral inequality

I need help with this exercise: let $f,g,h : [0,1] \to [0,\infty] $ integrable functions. Prove that the following statements are equivalent: i) $(f(x))^2 \leq g(x)h(x) $ almost everywhere. ...
2
votes
1answer
29 views

Justification of the use of Fubini's theorem in $\int_{(0,A)} \big[\sin(x) \int_{(0,\infty)}\exp(−nx)dn\big] \,dx.$

I am trying to figure the value of $$\lim_{A \to \infty} \int_{(0,A)} \frac{\sin(x)}{x} \,dx$$ using Fubini's theorem. As for now, I am thinking to substitute $1/x = \int_{(0,\infty)} \exp(−nx) \, ...
1
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0answers
32 views

What is the value of Lebesgue integral $\int_{E}{(f(x) - 1) dx}$?

I am trying to solve this problem: If $f : \mathbb{R} \to \mathbb{R}$ is a Lebesgue measurable function and $\int_{0}^{1}{f(x)dx} = 1$ (Lebesgue integral) and $E = \{x \in [0, 1] \mid f(x) > 1\}$, ...
3
votes
1answer
44 views

Convergence of series of integrals

Let $\phi \in C^\infty(\mathbb R)$ be a function such that $\phi(x), \phi'(x) \to 0$ as $x \to \infty$. I want to show that $$\lim_{n \to \infty} \int_\mathbb R \cos(nx) \phi(x) \ dx = 0$$ Doing it ...
1
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0answers
19 views

Computation of a Lebesgue-Stieltjes integral

I am asked to compute the integral $\int_{(0,3a]}x\,dF(x)$ with $a > 0$ where $$F(x) = \begin{cases} \pi & 0\leq x < a\\ 4+a-x & a\leq x < 2a \\ (x-2a)^2 & ...
2
votes
2answers
62 views

Continuity of a Lebesgue indefinite integral over unbounded interval

We know that if $f : [a,b] \rightarrow \mathbb{R}$ is Lebesgue-integrable, then $$ F(x) = \int_{a}^{x} f(t) dt $$ is continuous. But if $f : \mathbb{R} \rightarrow \mathbb{R}$ is ...
4
votes
1answer
43 views

Requesting hints for showing that some function that is locally $L^p$ integrable is in $L^1(\mathbb{R})$.

Suppose $\int _a^b\vert f\vert^p<\infty$ for some $p\ge 1$ and for all $a,b\in \mathbb{R}$, and for some $a>p-1$ $$\int_{2\vert y-x\vert \le x}\vert f(y)\vert ^pdy\le \vert x\vert^{-a}$$ when ...
-1
votes
1answer
39 views

Two Definitions of Lebesgue Integral

So the definition of Lebesgue integral as I understand it is as follows: Let $(X, \mathcal{F}, \mu)$ be a measure space, and $f: X \to [0, + \infty]$ a non-negative function. Then for simple ...
1
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1answer
38 views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
3
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0answers
31 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
1
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1answer
33 views

Complex Measures: Absolute Continuity [closed]

Note: This is a lemma for: Spectral Measures: Riemann-Lebesgue Given a positive measure: $$\lambda:\mathcal{A}\to[0,\infty]$$ Consider a complex measure: $$\mu:\mathcal{A}\to\mathbb{C}$$ How to ...
1
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1answer
34 views

Does the mean integral over B(x,r) of a L1 function u converge a.e. to u(x)?

Suppose $u\in L^1(\Omega )$. Let $u_{x,r}$ be the mean of $u$ over the ball $B(x,r)$ (s.t. $B(x,r) \subset \Omega$), i.e. $ u_{x,r} := \frac{1}{|B(x,r)|} \int_{B(x,r)} u(y) dy$. Is it true that ...
0
votes
0answers
31 views

Lebesgue integral of cartesian product of functions

Given two Lebesgue Integrable functions $f,g$, is there a notion of the integral $$\int_A f \times g \, \, dx_1 \times dx_2 ?$$ Is this even a definable notion? I couldn't find anything on the ...
1
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1answer
31 views

Double integral of $e^{3+y^2}$ over a triangle

Evaluate $\iint_{A}^{} e^{3+y^2}dxdy$ where $A$ is a triangle with vertices $(0,0)$, $(0,-1)$ and $(1,-1)$. I don't know how to bite that. I tried multiplying it by $e^{x^2}$ and then changing the ...
1
vote
1answer
38 views

Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$

I am reading Lebesgue Integration on Euclidean Space by Frank Jones. My question is specifically regarding Chapter 4, Section C titled "The Lebesgue Function Associated with a Cantor Set". The author ...
3
votes
2answers
101 views

Finding integral over inconvenient set

Put $F = \{ (x,y) \in \mathbb{R}^2 : |x^2-y^2| \leq 1, 2|xy| \leq 1 \}$. How do we find the following integral? $$\int_F (x^2 + y^2) \,d(x,y)$$ I'm sure we need to use Jacobi's transformation ...
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0answers
29 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
1
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1answer
29 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
3
votes
1answer
48 views

Computing with Lebesgue integrals

This problem comes from Royden's Real Analysis, 4th ed., pg 84, #19: For a number $\alpha$, define $f(x)=x^\alpha$ for $0<x\le 1$ and $f(0)=0$. Compute $\int_0^1 f$. MY WORK: I know ...
2
votes
1answer
19 views

Sequence of continuous functions convergent to $0$ with the integral equal to $1$

I am looking for a sequence of continuous functions $\{f_m\}$ defined in $A\subset\mathbb{R}$ with $\lim\limits_{m\to\infty} f_m=0$ such that $\int_A f_m \;d\mu=1$. The problem I have is with the ...
0
votes
1answer
14 views

Normaliztion in$L^{p}$ and $L^{q}$

Given a function f in $L^{p}$ and $L^{q}$ where $0<p,q<\infty$ Is f can always be normalized s.t. $\left\Vert f \right\Vert_p=\left\Vert f \right\Vert_q=1$
3
votes
1answer
54 views

Convergence of $f_n(x) = 2^n \cdot F(2^n (x-a_n))$ with $F(x) = e^{-x^2}$ with different notions of convergence.

I had my measure theory exam this morning, and one exercise was the following: I really can't see a solution. During the semester, we talked about almost everywhere convergence, almost uniform ...
2
votes
0answers
55 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...
2
votes
1answer
22 views

Non-negative Lebesgue Measurable Functions/Determining Measure of a Particular Set

I'm having some difficulties trying to figure out where to even start with this problem: Let $f$, $g$ be non-negative, measurable functions on $\left[ 0,1 \right]$ such that $\int_0^1 f(x)dx=2$, ...
1
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1answer
32 views

Show that $\mu$ is absolutely continuous w.r.t. $\mathcal{L}$ and find $\frac{d\mu} {d\mathcal{L}}$

Let $\mu$ be the unique Borel measure on $\mathbb{R}$ satisfying $\mu((a,b])=\arctan b-\arctan a$. Show that for any $\mu$-measurable subset $E$ of $\mathbb{R}$, $\mathcal{L}(E)=0$ implies ...
1
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1answer
54 views

Royden - section 4.2, page 73 - linearity

In Royden's "Real Analysis" on page 73, after the proof of linearity and monotonicity of the Lebesgue integral of simple functions, there's a little paragraph that says that this linearity shows that ...
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1answer
32 views

Solving this discontinuous integral using Lebesgue

Not a duplicate look at $f(x)$ here! Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is irrational}, & \newline 0 ...
9
votes
5answers
545 views

Evaluating Integrals using Lebesgue Integration

Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is rational}, & \newline 0 \space \text{if} \space x \space \text{is ...
0
votes
1answer
48 views

Prove that $\int_{[c,d]}|f(x,y)|d\mathcal{L}(y)<\infty$ for $\mathcal{L}$-almost all $x\in [a,b]$.

Suppose $f(x,y)$ is a Borel function on $\mathbb{R}^2$ which is in the $L^2$-space with respect to the $\mathcal{L}\times\mathcal{L}$. Prove the following: Given any finite rectangle ...
2
votes
2answers
39 views

Help with a Royden exercise of measure

I'm solving the exercise 12, of section 4 The General Lebesgue Integral from the Royden's book Real Analysis 3rd edition: Let $g$ be an integrable function on a set $E$ and suppose that $(f_n)$ is a ...
3
votes
1answer
50 views

Let A be a measurable set in R. Let B be all of it's densed points. is B necessarily open?

Let $A \subset \mathbb{R}$ be a measurable set. Define $B$: $$B =\left\{x\in \mathbb{R}: \lim \limits_{\epsilon \to 0^+} \frac{m([x-\epsilon, x+\epsilon]\bigcap A)}{2\epsilon} = 1\right\}$$ Is $B$ ...
1
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0answers
75 views

The difference between Riemann integrable function and Lebesgue integrable function

My professor asked my how to intuitively understand Lebegue Dominated Convergence Theorem and what's the effect of the integrable dominated function. More specifically. when we are given a Lebesgue ...
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0answers
51 views

Difficult question on integral

we denote by $\overline{u}$ a positif fuction "radially symmetric about the origin" that realize $$\inf\{\int_{\mathbb{R}^N} (|\nabla u|^2+\lambda u^2) dx, u\in H^1_0({\mathbb{R}^N}), ...
0
votes
1answer
28 views

how to prove this equality $||f||_{L^p}^{p}=p\int_0^{+\infty} \lambda^{p -1}\mu(E^f_\lambda) d\lambda$

Let $(X,B(X),\mu)$ be a measure space, suppose there is a function f that is measurable Define the distribution function ${\mu(E_\lambda^f): {\mathbb R}^+ \rightarrow [0,+\infty]}$ How to prove ...
1
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2answers
82 views

Minkowski inequality of infinite sum

For $1\leq p <\infty,$ Given $\{f_n\}^{\infty}_{n=1}$ be a sequence of function in $L^{p}(\mathbb{R}).$ Show that $\left\Vert \sum\limits_{n=1}^\infty f_n\right\Vert_p \leq \sum\limits_{n=1}^\infty ...
1
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1answer
29 views

Reference for integration theorem

I need a reference. In my demonstration there is this passage: $$\int_\Omega v(a-b)d\Omega = 0 \qquad \forall v \in V$$ With $a,b \in V$, a space that allows the integration (ex. $L^2(\Omega)$) ...
3
votes
2answers
52 views

Is Lebesgue integral over interior equal to the integral over the whole set?

I have a measurable set $S\subset\mathbb{R}$ and a measurable function $f\colon\mathbb{R}\rightarrow \mathbb{R}$. Is it true that $$\int\limits_Sf(x)\, dx=\int\limits_{\operatorname{int}(S)}f(x)\,d ...
0
votes
2answers
35 views

Find integral $\int_{R^2}\; \exp (-x^2-xy-y^2)\,dl_2(x,y)$

Find integral $$\int_{R^2} \exp (-x^2-xy-y^2)\,dl_2(x,y)$$ Should I use Fubini theorem and divide it into 2 separate integrals? Still not sure how to do it.
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1answer
41 views

Mollifiers: Asymptotic Convergence vs. Mean Convergence

Problem Does asymptotic convergence imply mean convergence: ...
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0answers
87 views

Mollifiers: Derivative

This thread is meant as lemma for: Semigroups & Generators: Entire Elements: Construction Given a smooth mollifier: $\varphi\in\mathcal{L}(\mathbb{R}): \varphi'\in\mathcal{L}(\mathbb{R})$ Do ...
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3answers
64 views

How do you find this limit $\lim_{n\to +\infty} \int _{\frac{1}{n}}^{n} \frac{|\sin x|^n}{x^{\alpha}}\,dx $

I don't know how to solve the limit $$\lim_{n\to +\infty} \int _{\frac{1}{n}}^{n} \frac{|\sin x|^n}{x^{\alpha}}\,dx $$ for each $\alpha>1$. My attempt: $\displaystyle ...
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0answers
47 views

Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
0
votes
1answer
56 views

Showing that $f$ is not Absolutely continuous

Frist:- I am not sure about what title this question should be. Suppose the function $f:[0,\frac{1}{2}]\rightarrow \mathbb{R}$ defined by $$ f(x) = \begin{cases} 0, & \text{if }x=0 \\ x ...
2
votes
2answers
71 views

Lebesgue integral and absolute value

I wonder why we say that $f$ is integrable iff $\int|f|\,d\mu$ is finite? Why we use absolute value? Won't it be enough to have that $\int f\, d\mu$ is finite to call $f$ integrable? Are there ...
6
votes
1answer
144 views

Showing that a function is in $L^1$

I need to prove the following statement or find a counter-example: Let $u\in L^1\cap C^2$ with $u''\in L^1$. Then $u'\in L^1$. Unfortunately, I have no idea how to prove or disprove it, since the ...
1
vote
1answer
40 views

Study the convergence of the sequence of functions $f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$ (convergence in measure, pointwise and in $ L^2(R ^d)$

Study the convergence of the sequence of functions $$f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$$ (convergence in measure, pointwise and in $ L^2(\mathbb{R} ^d)$). Let f be a measurable function such ...
0
votes
0answers
20 views

Why do we construct the Lebesgue measure with finite measure sets before sets of arbitrary measure? [duplicate]

On page 20 of the following lecture notes, Stage 5 constructs the Lebesgue measure on finite sets before constructing it on arbitrary sets as in Stage 6: ...
2
votes
0answers
39 views

Show that $d+1$-dimensional Lebesgue measure of set $G$ equals $0$

Let $D \subset \mathbb{R}^d$ and let $f:D \rightarrow \mathbb{R} $ be measurable function. Let $G=\{(x_1,x_2,\ldots,x_d,f(x_1,x_2,\ldots,x_d))\in \mathbb{R}^{d+1}:(x_1,x_2,\ldots,x_d)\in D \} $ be the ...
2
votes
0answers
48 views

Riemann and Lebesgue improper integral Proof

I've been trying to find some notes on the following statement: Let $f:(a,b] \to \mathbb{R}$, $f\geq 0$, and $f\in\mathcal{R}[a+\epsilon , b]$ for any $\epsilon>0$. Then $\int_a^bf=\lim_{\epsilon ...
3
votes
4answers
130 views

Computing $\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$

I want to compute $\int_0^\infty u^{-1}(1-e^{\frac{-u^2 t}{2}})\sin(u(|x|-r))\,du$ and so ,as shown below, I want to compute $$\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$$ Attempt We split ...