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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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
33 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 ...
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5answers
548 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 ...
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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 ...
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2answers
42 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 ...
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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 $B$ ...
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0answers
78 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}), ...
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1answer
29 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 ...
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2answers
89 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 ...
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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. $L^2(\Omega)$) ...
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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 ...
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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
88 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
66 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: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
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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 ...
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2answers
73 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 ...
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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 ...
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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 ...
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0answers
21 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: ...
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0answers
40 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
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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
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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 ...
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1answer
35 views

Borel-Stieltjes measure problem with floor functions.

We have 2 functions, $$\alpha(x)=\left\{ \begin{array} {cl} \lfloor x \rfloor + \log(1+x),& x\geq 0 \\ \lfloor x \rfloor, & x<0 \end{array} \right.$$ $$\beta(x)=\lfloor x \rfloor$$ ...
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1answer
24 views

Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$

Let $f$ integrable and let $E_\alpha=\{x\mid |f(x)|>\alpha\}$ for $\alpha\geq 0$. Show that $$\int\left|f(x)\right|dx=\int_0^\infty m(E_\alpha)d\alpha$$ where $m$ is the Lebegue measure. By ...
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1answer
35 views

weakly convergence imply strong convergence when $ \|f_n\| \rightarrow \|f\| $ in $l^2([0,1])$? [duplicate]

I know in general weakly convergence do not imply strong convergence in $L^p$,but in $L^2[0,1]$ space which if we have additional condition do this condition plus the weak convergence will give us ...
3
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1answer
81 views

A function $f$ such that $f \in L_1$ but $f \notin L_p$ for $p>1$ [duplicate]

I want find a function $f: [0,1] \mapsto \mathbb{R}$ such that $f \in L_1[0,1]$ but $f \notin L_p[0,1]$ for all $p>1$. My attempts: First I thought in the family of functions $\frac{1}{x^\alpha}$ ...
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1answer
20 views

$\iint |x-y|^{-t} \,d\mu\, d\mu < \infty$ iff $t<1$

Consider the measure space $([0,1], \mathcal{L}([0,1]), \mu)$, where $\mu$ is the restriction of the Lebesgue measure to the closed interval $[0,1]$. I wish to show $\iint |x-y|^{-t} \,d\mu\, d\mu ...
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1answer
34 views

Integral of a simple function

The definition of a simple function is that let ($\Omega$,F, $\mu$) be a measure space and for let $\Omega$ be written as disjoint union of $A_i$'s where $i=0,1,..,n$ . A function $f$ from $\Omega$ to ...
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0answers
43 views

Mixed joint density?

I would like to know how to get the following result : According to wikipedia : http://en.wikipedia.org/wiki/Joint_probability_distribution#Mixed_case We define the mixed joint density as ...
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0answers
37 views

how to show that $\int_A f \, d\mu =\mu(A) \int_X f \, d\mu$ when $\mu(E\cap A)=\mu(E) \mu(A)$

Suppose $(X,M,\mu)$ is a measure space and there exists a set $A\in M$ with $\mu(A)<\infty$ such that $\mu(E\cap A)=\mu(E) \mu(A)$ of all $E\in M$. I want to show that $\int_A f \, d\mu =\mu(A) ...
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3answers
82 views

Distributions defined by $C_0^\infty(\mathbb{R})$ enough to distinguish $f_1,f_2\in L^1(\mathbb{R})$?

Let $f_1,f_2$ be Lebesgue-summable functions on the real line. I was wondering whether space $C_0^\infty(\mathbb{R})$ of infinitely differentiable compactly supported functions, intended as ...
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4answers
75 views

Example where $\lim\limits_{m\rightarrow\infty} \int_E f_m =0$ and $f_m(x) \not\rightarrow 0$

I am looking for an example of a sequences of non-negative and measurables functions with $\lim\limits_{m\rightarrow\infty} \int_E f_m =0$ and $f_m \nrightarrow 0 \:\:\forall\:x\in E$
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1answer
36 views

Is $\left(\frac{1}{x}\right)^{\frac{1}{p} - \frac{1}{n}}$ a Cauchy Sequence in $L^p((0,1))$

Is $(\left(\frac{1}{x}\right)^{\frac{1}{p} - \frac{1}{n}})_{n\in N}$ a Cauchy Sequence in $L^p((0,1))$? and does it converge to $\frac{1}{x}^{\frac{1}{p}}$ (p is a real number bigger or equal to 1) I ...
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3answers
60 views

Let $0<a<b$. Calculate $\int_{(0,1)}\frac{t^b-t^a}{\ln(t)}dt$.

Assignment: Let $0<a<b$. Calculate $$\int_{(0,1)}\frac{t^b-t^a}{\ln(t)}dt$$ I'd appreciate a little help with this one. A hint says that rewriting $t^b-t^a$ as an integral should help, but ...
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1answer
42 views

Prove that $C^\infty(\mathbb{R}^n)$ is dense in $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d\xi$

I would like to show that $C^\infty(\mathbb{R}^n)$ is dense in the space $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d \xi)$ (here, $s$ is an arbitrary element of $\mathbb{R}$). I am familiar with the ...
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0answers
30 views

zeros of the dyadic maximal function

Recall the definition of the Hardy-Littlewood maximal function $Mf$ (https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function). If we replace the balls in the definition by dyadic cubes ...
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1answer
38 views

Relation between two answered problem in Lebesuge Integral

yesterday I asked a question and we get the answer, for reference this is what we ask Problem 1 : Let $(X,M,\mu)$ be a measure space and $f$ is a real-valued function on $X$ such that $$\int_X |f| ...
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1answer
63 views

If $f$ is Lebesgue integrable, then for every $\epsilon>0$ there is a set $E$ of finite measure such that $\int_{E^c}|f|<\epsilon$

If we have a measure space $(X,M,\mu)$ and $f$ is a real function on $X$ such that $$\int_X |f| d\mu <\infty$$ ( in other word $f$ integrable). How to prove that for any $\epsilon >0$ we can ...
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1answer
43 views

Uniform continuity of the antiderivative

We know that if $f:\mathbb{R}\to\mathbb{R}$ is a function such that $$\sup_{x\in\mathbb{R}}|f(y)|<\infty,$$ then the function $g(x)=\int_0^xf(y)dy$ is uniformly continuous. I am just wondering ...
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0answers
24 views

Borel function and characteristic equation

Define a Borel probability measure $\mu_n $ by $\mu_n ({x}) = \frac{1}{n} $ for $x = 0, \frac{1}{n}, \frac{2}{n}, ..., 1-\frac{1}{n} $. Let $\eta$ be a Lebesgue measure on $[0,1]$. i) I'm to compute ...
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1answer
134 views

Expectation of a discrete random variable: how to convert an integral to a sum?

According to wikipedia and all my textbooks, we define the expectation of a random variable on a probability space $(\Omega, \mathcal{F},P)$ : \begin{align} E(X) &= \int_{\Omega}XdP\\ \end{align} ...
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1answer
45 views

Conditions on integration by parts with unbounded endpoint

I have the following theorem for integration by parts when both endpoints are finite: (Lebesgue integrals are used throughout) Let $a\le b$ be real numbers, and $f,g$ be functions continuous on ...
2
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3answers
89 views

integral of Lebesgue Measurable function defined on finite measure.

Let $f(x)$ be a nonnegative Lebesgue measurable function on $[a,b]$ and let $E_n=\{x : f(x) \ge n \}$. How to prove that $f$ is integrable if and only if $$ \sum_{n=1}^{\infty} \mu(E_n)<\infty.$$ ...
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1answer
106 views

Evaluate the limit integral using the Lebesgue Dominated Convergence Theorem

I have tried to use the Lebesgue Dominated Convergence Theorem to evaluate: $$\lim_{n\rightarrow \infty} \int_{(0,1]} f_n \;d\mu $$ with $f_n(x)=\dfrac{n\sqrt{x}}{1+n^2x^2}$ and ...
3
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2answers
86 views

Integrate $\int_0^\infty \int_0^\infty \frac{\sin \pi x}{(y+e^x|\sin \pi x|)^2}dx \, dy$ using Fubini or Tonelli theorems

I am trying to show that this integral $$\int_0^\infty \int_0^\infty \frac{\sin \pi x}{(y+e^x|\sin \pi x|)^2}dx \, dy $$ exists and is finite and then finding its value. Since $\sin \pi x$ takes ...
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1answer
25 views

Lebesgue integral, integer part x

$$ \int_{0}^{\infty} 10^{-2[x]} dx $$ How to solve it? is the Lebesgue integral. I drew a graph, it is piecewise continuous. Sum of this function will converge. But I can not understand how it all ...
5
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0answers
78 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, ...