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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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
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1answer
49 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 $L^2(0,1)$, ...
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1answer
43 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 ...
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2answers
29 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
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2answers
34 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 ...
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2answers
42 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 ...
3
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0answers
41 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 ...
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1answer
38 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 ...
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3answers
37 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 ...
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2answers
49 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
27 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 ...
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0answers
12 views

positive integrable part implies downside integrable

Let $A: M\rightarrow GL(d)$ measurable where $(M, \mathcal{B},\mu)$ is a probability space, then are equivalent: $$\log^+\Vert A^{\pm1}(x)\Vert\in L^1(\mu)\Leftrightarrow \log^-\Vert ...
3
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1answer
42 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 $0<p<2$, ...
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2answers
27 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
32 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: ...
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1answer
83 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. ...
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1answer
20 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
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1answer
45 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 $p=4$.
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1answer
18 views

Limit of integral of Lebesgue integrable function [closed]

Let $f:\Bbb{R}\to\Bbb{R}$ be a Lebesgue integrable function. Is $ \lim\limits_{n\to\infty}\int_{\lvert x\rvert\geqslant n} f(x) dx = 0 ?$
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1answer
33 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. ...
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1answer
13 views

Lebesgue integral of cardinal

What is $ \int_{E} f d\mu $ if $\mu = \mu_c$ the cardinal measure on $\mathbb{N}$.
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1answer
36 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 ...
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1answer
98 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
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2answers
56 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?
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1answer
84 views
+50

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 ...
2
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1answer
56 views

If $\int_A f=0$ for every measurable subset $A$ of $E$, then $f(x)=0$ a.e. on $E$

Is there a function $f$ that doesn't satisfy in the follwing statement? If $\int_A f=0$ for every measurable subset $A$ of $E$, then $f(x)=0$ a.e. on $E$.
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1answer
33 views

calculate integral using lebesgue dominated convergence theorem

Can anyone give me a hand on how to calculate the following using lebesgue dominated convergence theorem: lim n→∞∫[0,1] (n(sin(x/n)))^n dx I wonder if as n approach infinity, (n(sin(x/n)))^n become ...
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2answers
46 views

How to use dominated convergence theorem?

How to use dominated convergence theorem to compute $$lim_{n\rightarrow \infty}\int_0^1\frac{1+nx^2}{(1+x^2)^n}$$ So far I have only done $\frac{1+nx^2}{(1+x^2)^n}\le\frac{1+nx^2}{(1+x^2)}$, I don't ...
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2answers
40 views

Lebesgue Integration Formula Help

I have been endeavouring to teach myself Lebesgue's method of integration, and while theory has been okay, I have had some severe difficulty with the practical side (most texts seem to not provide a ...
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2answers
32 views

Integrability: Cauchy Sequence

This thread is related to: Spectral Measure: Dominated Convergence Given a measure space $\Omega$. Consider a sequence of square integrables: $\int|f_n|^2\mathrm{d}\mu<\infty$ Suppose pointwise ...
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2answers
45 views

Help with a tricky limit $\lim_{n\to\infty} \sum\limits_{i=1}^n (i/n)(\sqrt[2]{(i+1)/n}-\sqrt[2]{i/n})$

I have been attempting to follow the answer to a question previously asked on the site (Lebesgue integral basics), but am lost on how one might go about evaluating $$\lim_{n\to\infty} ...
3
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1answer
49 views

Increasing functions on $\mathbb{R}$

If $F$ is increasing on $\mathbb{R}$ then show that $F(b)-F(a)\geq \int_a^b F'(t)dt$. My work: Since $F$ is increasing on $\mathbb{R}$, $F'$ exists a.e. on $\mathbb{R}$. So $F'$ is integrable on ...
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2answers
28 views

If $f_n \rightarrow 0$ and $\int \sup f_1, … , f_n \leq M$ then $\int f_n \rightarrow 0$

Let $f_n$ be a sequence of nonnegative measurable functions which converge to $0$. If there exists an $M$ such that $$\int \sup f_1, ... , f_n \leq M$$ for all $n$, then $\lim \int f_n = 0$. Could ...
4
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2answers
41 views

Prove the following inequality: $\int_{(a,b)}f\ d\lambda\cdot\int_{(a,b)}\frac{1}{f}d\lambda≥(b-a)^2$

Assignment: Let $-\infty < a < b < \infty$ and $f: (a,b) \rightarrow (0,\infty)$ be measurable, such that $f$ and $\frac{1}{f}$ are Lebesgue integrable. Prove the following inequality: ...
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1answer
21 views

Existence everywhere of integrand in Fubini's theorem

Let $f\in L(A,\mu_x\otimes\mu_y)$ be a summable function on $A\subset X\times Y$ where $(X\times Y,\mu_x\otimes\mu_y)$ is the product of measure spaces $(X,\mu_x)$ and $(Y,\mu_y)$. Then Fubini's ...
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1answer
24 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
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1answer
26 views

an inequality on $L_p$ and $l_2$

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that ...
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0answers
14 views

Definition of Measure regular

Book's, Real and complex analysis, Walter Rudin. I am somewhat confused. My question is: "In other words, we are looking at $L^p$, where $\mu$ is Lebesgue measure on $[0,2\pi]$(or on $T$), ...
2
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1answer
50 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
2
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2answers
43 views

Confused about substitution in Stiltjes integral

Suppose we have an integral $$ \int_{-a}^{a} \sin (x) \nu(dx), $$ where $\nu$ is a finite measure with $\nu(-A)=\nu(A), A \in \sigma(\mathbb{R})$ and $x>0$. Then we have $$ \int_{-a}^{a} \sin (x) ...
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1answer
19 views

What does it mean to be absolutely integrable on $\mathbb{R}$ and what are the steps to show that something is absolutely integrable?

I just have a quick question. What does it mean to be absolutely integrable on $\mathbb{R}$ and what are the steps to show that something is absolutely integrable? For example what if we wanted to ...
3
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1answer
51 views

Evaluation of Lebesgue Integral using Convergence Theorems

Using convergence theorems, I am trying to compute the value of $$ \lim_{n\to\infty}\int_a^\infty \frac n{1+n^2x^2}\,\mathbb{d}x $$ for $a \in \mathbb{R}$, and with respect to the Lebesgue measure. ...
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1answer
23 views

Bochner Integral: Approximability

Disclaimer This thread is related to: Bochner Integral: Integrability It is meant to record. See: Answer own Question It is written as jeopardy. Have fun! :) Problem Given a measure space ...
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1answer
25 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
2
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0answers
23 views

Proving translational invariance of Lebesgue integral

I am asked to show that the Lebesgue integral is invariant under translations. Specifically, Let $(\mathbb{R}, \Sigma, \mu)$ be a measure space, and for any $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
1
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1answer
20 views

Degenerate Hilbert-Schmidt operators

Let us define a Hilbert Schmidt operator $A:L_2[a,b]\to L_2[a,b]$ by $$A\varphi:=\int_{[a,b]} K(s,t)\varphi(t)d\mu_t$$where $\mu_t$ is the linear Lebesgue measure. A degenerate case is represented by ...
1
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2answers
80 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
0
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0answers
23 views

Derivative of $g(t) = \int_0^t u(s)$ for $u \in L^2(0,T)$.

Let $u \in L^2(0,T)$. Consider $g(t) = \int_0^t u(s)\;ds$. Is it true that $g'(t) = u(t)$ a.e? How would I show this? Does some stronger statement hold?
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1answer
28 views

Measurability of function defined by an integral

Let $A$ be a Hilbert-Schmidt operator defined on $L_2[a,b]$ by $$A(\varphi):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$where $K\in L_2([a,b]^2)$. The fact that $A(\varphi)\in L_2[a,b]$ is showed in the ...
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1answer
17 views

Properties of function on $L_p$ spaces

Given $L_p$ space with the lebesgue measure on $\mathbb{R}^n$ and the function $f(x) = |x|^{-\alpha}$ if $|x| < 1$ $f(x) = 0$ if $|x| \geq 1$ I need to show that $f \in L_p$ if and only if ...