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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1answer
25 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 ...
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1answer
25 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
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2answers
83 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
13 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 ...
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1answer
24 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
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1answer
30 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 ...
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1answer
17 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 ...
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1answer
13 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
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1answer
43 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 ...
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0answers
37 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
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1answer
20 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
27 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 ...
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1answer
37 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
31 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
492 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
44 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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1answer
40 views

A function such that $\int f$ is convergent and $f$ is not integrable [closed]

Could you give me an example of a function such that $\int f$ is convergent and $f$ is not integrable ? It look very strange to me, it was a question at my exam yesterday.
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2answers
26 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
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1answer
41 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
53 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
50 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
23 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
41 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
28 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
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2answers
43 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
30 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
34 views

Mollifiers: Asymptotic Convergence vs. Mean Convergence

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

Mollifiers: Derivative

Problem Given a smooth mollifier: $\varphi\in\mathcal{L}(\mathbb{R}): \varphi'\in\mathcal{L}(\mathbb{R})$ Do the derivatives exist in the sense: ...
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3answers
55 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
32 views

Semigroups & Generators: Entire Elements: Construction

Problem Given a Banach space $E$. Consider a $\mathcal{C}_0$-group(!): $T:\mathbb{R}\to\mathcal{B}(E)$. Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)$$ (The domain being those ...
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1answer
48 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
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2answers
53 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
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1answer
121 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
38 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
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: ...
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0answers
31 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
41 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 ...
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4answers
117 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
30 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
23 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
26 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
61 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 $\frac{1}{x^\alpha}$ ...
2
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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 ...
1
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1answer
26 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
18 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 ...
1
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0answers
33 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) ...
0
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3answers
80 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
73 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 $f_m \nrightarrow 0 \:\:\forall\:x\in E$
1
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1answer
32 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 ...
1
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3answers
58 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 ...