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
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
62 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 ...
0
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0answers
19 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
34 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 ...
1
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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
59 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 ...
2
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1answer
41 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 ...
2
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0answers
23 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 ...
1
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1answer
37 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| ...
3
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1answer
56 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 ...
0
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2answers
35 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 ...
1
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0answers
22 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 ...
0
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1answer
81 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} ...
1
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1answer
40 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
66 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 $$ \sum_{n=1}^{\infty} \mu(E_n)<\infty.$$ ...
-1
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1answer
74 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
71 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 ...
0
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1answer
24 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
67 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, ...
1
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1answer
27 views

Determining the orthogonal complement of $\{1 \}^\perp$ in $L^2[0,1]$

Consider the space $L^2[0,1]$ of complex valued square-integrable functions $f : [0,1] \to \mathbb{C}$. Let $\langle f, g \rangle = \int_0^1 f \bar{g}$ denote the standard $L^2$ inner product. For $M ...
2
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0answers
36 views

Prove that $\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}$ [duplicate]

Let $(X,\mathcal{A}, \mu)$ be any measure space and let $u \in \bigcap_{p\in [1,\infty]} \mathcal{L}^p(\mu)$. Then $$\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}.$$ I have already proved the ...
1
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1answer
39 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
2
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0answers
45 views

Lebesgue integral over $\mathbb R^2$ of the function $f(x,y)=2(x-y)e^{-(x-y)^2}\chi_{\{x>0\}}$

Let $f:\mathbb R^2\to \mathbb R$ be given by $$f(x,y)=\begin{cases}2(x-y)e^{-(x-y)^2}& \text{ if }x>0 \\0&\text{ otherwise}\end{cases}$$ Given that $\int^\infty_{-\infty} e^{-z^2} dz=\sqrt ...
1
vote
1answer
56 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$

Could you please help me solving this old prelim problem. Any hints are appreciated
0
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1answer
56 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
1
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1answer
49 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
0
votes
2answers
41 views

Show $\sup_{y>0}\left|\int_0^\infty \int_t^\infty f(x,y) \cos\left(\dfrac{t}{y}\right)dx\,\,dt\right|<\infty$

Suppose $f$ is Lebesgue measurable on $[0,\infty)\times [0,\infty)$ and $g\in L^1([0,\infty))$. If $|xf(x,y)|\leq g(x)$ for all $y\in [0,\infty)$ prove that $$\sup_{y>0}\left|\int_0^\infty ...
2
votes
2answers
36 views

Finding the limit of this integral: $\lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$ if $q<p+1$

I am trying to find the following limit provided: $q<p+1$: $$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$ Dividing by $n x^q$ so we have $$\dfrac{n x^p+x^q}{x^p+n ...
0
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2answers
44 views

Why is $E[X1_A]=0$ if $P(A)=0$?

I know this is trivial and intuitive, but I'm not able to convince myself rigorously. If $P(A)=0$, why is it true that $E(X1_A)=0$? Every book discards it out as an obvious fact. I tried to prove it ...
4
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0answers
107 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
1
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1answer
43 views

If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$

Problem: If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$ in $L^p$. An official solution I saw for this problem looked very different. Here is my ...
3
votes
3answers
40 views

$\int_x^z f=0$ for every $z\in[x,y]$, then $f(a)=0$ a.e. on $[x,y]$.

I have to prove the following statement Let $f$ be bounded measurable function on$[x,y]$. Suppose that $\int_x^z f=0$ for every $z\in[x,y]$, then $f(a)=0$ a.e. on $[x,y]$. I suppose that $f$ is not ...
4
votes
2answers
61 views

If $\int_E f=\int_E g$ then $f=g$ a.e.?

Is the converse of the following statement is true? Let $f$ and $g$ be two bounded measurable functions on a set $E$. If $f(x)=g(x)$ a.e. on $E$ then $$\int_E f=\int_E g$$ Here is my proof for ...
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2answers
57 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
3
votes
1answer
44 views

If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

Let $M>0$, $\{f_n\}\subset L^2([0,1])$ such that $\int_0^1 |f_n|^2 dm\leq M$ and $f_n(x)\to 0$ as $n\to\infty$ almost everywhere, $m$ is Lebesgue measure. Show that for all $0<p<2$, ...
1
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2answers
33 views

Using the MCT to evaluate the integral of a series

I'm studying for my Measure Theory final and I've come across a question that I can't seem to find an answer for. For each $n \in \mathbb{N}$ set $E_n:=[n,2n]$ and let $f:\mathbb{R} \to \mathbb{R}$ ...
2
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1answer
35 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
2
votes
1answer
97 views

$\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$ implies $\lim_{n \to \infty} \int_B f_n \, d\mu = \int_B f \, d\mu$ for $B \subseteq X$

I'm having trouble with the following problem. Let $(X, \mathcal{M},\mu)$ be a measure space, where $X = [a,b] \subset \mathbb{R}$ is a closed and bounded interval and $\mu$ is the Lebesgue measure. ...
1
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1answer
21 views

control of an integral using maximal function

Let $I$ be a compact interval with center $c(I)$ and N be a large positive integer. It seems to me that there exists a constant $C$ such that for any good function $f$ (e.g. Schwartz function) we have ...
3
votes
1answer
54 views

Find a function in $L^p(\mathbb{R})$ only for $p=4$ [duplicate]

I'm having trouble with this problem from an old analysis qual: Find a function $f$ such that for $p\in (1,\infty)$, $f$ is in $L^p(\mathbb{R})$ only when $p=4$.
1
vote
1answer
34 views

Lebesgue integrable function in rationals

Function $f : [0,1] \to \mathbb{R}$ defined as $ f(x) = \begin{cases} 1 & x\notin\mathbb{Q}\\ 0 & x\in\mathbb{Q} \end{cases} . $ As is well known $f$ is not integrable in the Riemann sense. ...
0
votes
1answer
14 views

Lebesgue integral of cardinal

What is $ \int_{E} f d\mu $ if $\mu = \mu_c$ the cardinal measure on $\mathbb{N}$.
1
vote
1answer
40 views

Convergence of series by using counting measure

Problem; Let $\{a_n\}$ and $\{r_n\}$ be two sequences of real numbers such that $\displaystyle\sum_{n\geq 1} |a_n|<\infty$. Prove that $$\sum_{n\geq 1} \frac{a_n}{\sqrt{|x-r_n|}}$$ converges ...
1
vote
1answer
104 views

Fatou: Reverse?

Attention The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$ (There Fatou may help out!) But as proceeding with Fatou one encounters ...
3
votes
2answers
67 views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
2
votes
1answer
108 views

If a joint cdf is increasing in each argument, then the pdf is strictly positive a.s.?

Let $F:\mathbb{R}^d \to [0,1]$ be an absolutely continuous joint cdf and let it be strictly increasing in each argument. Does it imply that its pdf $f$ is strictly positive a.s. (with respect to the ...