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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26 views

Let f be a measurable and finite a. e. on [0, 1]. prove that f=0 a.e. on [0, 1]. [duplicate]

Let f be a measurable and finite a. e. on [0, 1] and if $ \int_{E} f = 0 $ for all measurable set $ E \subset [0,1] $ with m(E)= 1/2, Then prove that f=0 a.e. on [0, 1]. If f is either non negative ...
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0answers
22 views

Interchanging Malliavin derivative with Lebesgue integral

I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the ...
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1answer
39 views

If $f$ is non negative measurable, $ \int f \lt \infty $ if and only if $\sum_{n=-\infty}^{\infty} 2^{n} m{(f\gt 2^n)} \lt\infty $

Let $f$ be non negative measurable. Prove that $ \int f \lt \infty $ if and only if $\sum_{n=-\infty}^{\infty} 2^{n} m{(f\gt 2^n)} \lt\infty $. This is a very popular question in Lebesgue ...
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2answers
70 views

Does there exist a sequence with $ \int_{[0,1]^2}| f_i - f_j |\, {\rm d}x = {\rm const.} > 0 $ for $ \forall i \neq j $?

Let $A = [0,1]^2$. Is there a such a sequence of functions $ f_i\colon A \rightarrow \mathbb{R} $ with $ \int_{A}| f_i | \ {\rm d}x \leqslant 1$ and $ \int_{A} | f_i - f_j | \ {\rm d}x = {\rm ...
4
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0answers
64 views

Represent total variation of continuous function by integration of counting function

$f : [a,b] \to \mathbb R$ is continuous, let $M(y)$ be the number of points $x$ in $[a,b]$ such that $f(x)=y$. prove that $M$ is Borel masurable and $\int M(y)dy$ equals the total variation of $f$ on ...
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1answer
24 views

Question for dominated convergence theorem.

Let $\{g_k\}$ and $g$ be integrable functions, $\{f_k\}$ and $f$ measurable functions, and $|f_k|\le g_k$, $f_k\to f$ almost everywhere If $$\lim_{k\to\infty} \int g_k\ \mathsf d\mu=\int g\ \mathsf ...
2
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0answers
18 views

lebesgue integral of S [duplicate]

Let $S$ be a bounded measurable subset of $\mathbb R$. Let $f \colon S → (0,\infty)$ be Lebesgue integrable. Prove that $$\lim_{n\to\infty}\int_S\ f^{1/n} \;\mathrm{d}m = m(S)$$ Where $m(S)$ is ...
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1answer
30 views

Prove that lebesgue integrable equal lebesgue measure

Let $S$ be a bounded measurable subset of $\mathbb R$. Let $f \colon S → (0,\infty)$ be Lebesgue integrable. Prove that $$\lim_{n\to\infty}\int_S\ f^{1/n} \;\mathrm{d}m = m(S)$$ Where $m(S)$ is ...
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1answer
17 views

$E$ measurable in $\mathbb{R}^2$ but $E^y$ not measurable for $y=0$

In the book of real analysis by Prof. Stein on p.76, I am confused about the following: Why is the set $E$ in $\mathbb{R}^2$ has measure zero and $E^y$ is ...
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0answers
36 views

Completion of $C^n([a,b])$ with respect to $L^2$ norm

I was wondering what is the completion of $C^n([a,b])$ under $L^2$ norm? Is it the whole $L^2$ space or just a dense subset of $L^2$?
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0answers
62 views

Why is this countable union of closed sets closed?

I'm having trouble understanding the logic of the following statement (taken from pg 53 of Lebesgue Integration on Euclidean Space by Frank Jones): "Though unions of countably many closed sets are ...
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1answer
44 views

Proof that $\lim_{n\to\infty}\int_S\ f^{1/n} \, dm = m(S)$

Let $S$ be a bounded measurable subset of $\mathbb R$. Let $f \colon S → (0,\infty)$ be Lebesgue integrable. Prove that $$\lim_{n\to\infty}\int_S\ f^{1/n} \;\mathrm{d}m = m(S)$$ Where $m(S)$ is ...
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1answer
137 views

when composition of continuous and Lebesgue integrable function Lebesgue integrable [closed]

Suppose $g:[a,b]\to\mathbb R$ is Lebesgue-integrable and $f:\mathbb R\to\mathbb R$ is continuous, then $f\circ g$ Lebesgue-integrable if $|f(x)|<a+b|x|$ for constants $a$ and $b$. How to prove if ...
2
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1answer
87 views

Show that $f$ is integrable $\implies $ $f$ finite a.e. [duplicate]

Show that $f:\mathbb R^d\longrightarrow \mathbb R$ is integrable $\implies $ $f$ finite a.e. My attempts I wanted to use Borel-Cantelli lemma, but my construction doesn't look good for it since I did ...
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0answers
24 views

Liminf/sup and integral with discontinuous integrand

Question I'm gathering information about estimate between liminf/sup and integral with discontinuous integrand. A typical setting on my mind is as follows: let $f:[0,1]\times[0,1]\to\mathbb{R}$ be a ...
2
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1answer
54 views

Convergence in $L^p$ spaces

Let $f_{n} \subseteq L^{p}(X, \mu)$, $1 < p < \infty$, which converge almost everywhere to a function $f$ in $L^{p}(X, \mu)$ and suppose that there is a constant $M$ such that ...
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1answer
82 views

Injectivity of Fourier transform between $L^1(\mathbb{R})$ and $C_0(\mathbb{R})$

The Fourier transform maps from $L^1(\mathbb{R})$ to $C_0(\mathbb{R})$ where $C_0(\mathbb{R})$ is all continuous functions that vanish as $x \rightarrow \infty$. Now given $f,g \in L^1(\mathbb{R})$, ...
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1answer
52 views

How to prove f=0 almost everywhere in this case?

The Lebesgue integrable function $f$ on $[a,b]$ satisfies the following condition: $\int_a^xf d\mu=0$ for any $x \in [a,b]$ I want to prove that $f=0$ a.e., so I tried $f=f_+-f_-$ , where $f_+=\max ...
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0answers
16 views

A property of Lebesgue integrable function

Let f be a nonnegative integrable function on measureable space (X,v). Then tv ({x: f (x)>t}) converges to 0, as t goes infinity. I want to prove this statement. I got that v ({x: f (x)>t}) ...
2
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0answers
32 views

Mathematical expectation of $F_\xi(\xi)$

Consider $F$ as a distribution function of some random variable $\xi$. The problem I'm trying to solve is to find integral: $$ \int_{-\infty}^{+\infty}F(x)dF(x) $$ From what I see, there are two ways ...
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1answer
67 views

Dominated convergence theorem (computing limit)

I need to compute $\displaystyle\lim_{n \rightarrow \infty} \int \frac{\sin (x^n)}{x^2} \, dx$ using Dominated Convergence theorem. I have taken the function $g$ such that $|f_n| \leq g$ , where $f_n ...
0
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1answer
29 views

Proving that a function is L1

Suppose $f \in L^1([0,b])$ and $g(x)=\int_x^b{\frac{f(t)}{t}dt}$ , prove that $g\in L^1([0,b])$ and $\int_{0}^{b} g(x) dx = \int_{0}^{b} f(t) dt$. Assume we are not allowed to use integration by ...
2
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0answers
40 views

Averages of integral and$ L^p$ space problem

Let $f: \mathbb R \to \mathbb R$ be an integrable function, for each $h>0$ let $$f_h(t)=\dfrac{1}{h}\int_{t-\frac{h}{2}}^{t+\frac{h}{2}}f(x)dx$$ Suppose $f \in L^P$, prove the following (1) $f_h ...
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2answers
62 views

Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$.

Assume $E$ has finite measure and $1 \leq p < \infty$. Suppose $\{ f_n\}$ is a sequence of measurable functions that converges pointwise a.e. on $E$ to $f$. For $1 \leq p < \infty$, show that ...
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0answers
67 views

$f$ locally bounded, nonnegative, and measurable function integrable iff series $\int_{n=1}^{\infty}a_{n}$ converges absolutely

Suppose $f$ is a locally bounded, nonnegative, and measurable function on $[1,\infty)$ and define $\displaystyle \int_{n}^{n+1}f$, $\,\,\forall n \in \mathbb{N}$. Then, is it true that $f$ is ...
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1answer
48 views

Lebesgue integral of non-negative

Assume that $f: [0,1] \rightarrow [0,\infty)$ is a Lebesgue measureable function such that $f(x) > 0$ for a.e $x.$. Show that for every $\epsilon >0$ there is $\delta >0$ such that for every ...
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0answers
22 views

Calculate Norm Operator

I'm trying to solve this exercice: Let $\omega(y)=y^{-4}$ and $L^{1}(\mathbb{R},\omega)$ the space of measurable functions $g:\mathbb{R}\rightarrow\mathbb{R}$ so that $g\omega$ is Lebesgue ...
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4answers
75 views

Show that the function is not Lebesgue Integrable

Related to this question: How do I show that $f(x)=\frac{(−1)^{n}}{n}$ for every $n⩽x<n+$ and $n\geq 1$ is not Lebesgue Integrable? I'm extremely confused by the notation. What I need to show is ...
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1answer
31 views

Show that measure of symmetric difference $= \int_{E}\vert \chi_{A} - \chi_{B}\vert$

I have just proven that the Nikodym (pseudo)metric $\rho(A,B)$ is a pseudometric (i.e., satisfies all the metric axioms, except that $\rho(A,B)$ can be $0$ even if $A \neq B$), and now I need to show ...
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0answers
21 views

Unbounded Case of $\int_{[0,1]}x^{\alpha}$

For a number $\alpha \in \mathbb{R}$, define $f(x) = x^{\alpha}$ for $0 < x \leq 1$ and $f(0) = 0$. I am tasked with computing $\int_{[0,1]} f$ in both the bounded and unbounded cases. I assume ...
2
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0answers
51 views

Change of variable for integration with respect to Haar measure

I know how to estimate the integral \begin{gather} \int f(Ub)\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{gather} where $f:S^n(\mathbb{R})\to \mathbb{R}$ ...
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2answers
44 views

The convolution of an integrable function with a $p$-integrable function is integrable

Let $\Sigma$ denote the set of Lebesgue-measurable subsets of $\mathbb{R}$, and $m$ the Lebesgue measure on $\mathbb{R}$. Let $1<p\leq \infty$, $f\in L^1(\mathbb{R},\Sigma,m)$, and $g\in ...
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1answer
50 views

Simple function with same support as a function with finite support

I am looking at a solution to the first part of problem 21 in Section 4.3 of Royden's Real Analysis. The part of the problem I am interested in states as follows: Let the function $f$be ...
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1answer
32 views

Measure of boundary in $\mathbb R^n$

I saw that the measure of the boundary of a regular open set in $\mathbb{R}^n$ is zero, so how can we talk about the integral on this boundary (for me it must be equal to zero always )? I saw that in ...
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1answer
37 views

Prove that if E is positive lebesgue measurabel set, then E − E and E + E contain non-empty open sets.

let E + E = {x + y : x, y ∈ E}, and define E − E similarly. Show that if E is a measurable subset of R of positive Lebesgue measure then E − E and E + E contain non-empty open sets. I have seen the ...
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1answer
50 views

Fredholm integral operator

Let $T(f)(x):=\int_{\mathbb{R}^d} k(x,y)f(y) dy$ where $k_1(x)= \int_{\mathbb{R}^d} |k(x,y)|dy$ and $k_2(y)= \int_{\mathbb{R}^d} |k(x,y)|dx$ are two $L^{\infty}$ functions. Then I want to show that ...
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0answers
29 views

Can we use a series of properties to determine integral operator $f \to \int_0^1 f d\mu $

Question: Suppose there exists an operator $I: C^{\infty}(0,1) \to \mathbb R$ satisfying the following properties: (1) $I (\chi_{(0,1)})=1$ ; (2) $I(kf)=kI(f)$, where $k\in \mathbb R$ and $f\in ...
5
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3answers
174 views

If $\int_A f\,dm = 0$ for all $A$ having some fixed measure $C$, then $f = 0$ almost everywhere

Let $ f \in L^1[0,1]$. Assume that there is a constant C, with $0 < C < 1$, such that for every measurable set $A \subset [0,1] $ with $m(A)=C$, we have $ \int_{A} f dm = 0 $. Prove that ...
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0answers
83 views

Problems with Proof of Jensen's Inequality (Durrett's “Probability Theory and Examples”)

I have some questions concerning the proof of the Jensen's Inequality I found in Durrett's "Probability Theory and Examples" [pp.23-24]. In the following there is the proof, with the questions I have ...
2
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1answer
46 views

Bounded variation and Integration

This is an exercise in section 6.3 of Royden & Fitzpatrick's Real Analysis. Consider the function $$f(x)= \begin{cases} x^a\sin\left(\frac{1}{x^b}\right) & \text{if }0 < x \leq 1 \\ 0 ...
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0answers
27 views

Difference of increasing functions differentiable a.e.

I'm working through Royden & Fitzpatrick's Real Analysis, in the beginning of section 6.3 it reads and I quote: "Lebesgue's theorem tells us that a monotone function on an open interval is ...
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2answers
83 views

Use the definition of the integral to prove that if $A \subset E$, where $E$ is measurable, then $\int_A f = \int_E f \chi_{A}$

I want to prove: If $A \subset E$, where $E$ is measurable, then $\int_A f = \int_E f \chi_{A}$, where $f$ is a bounded measurable function and $m(E) < \infty$ Solution: Let $f: E \to ...
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0answers
34 views

Expectation and the Survival Function: Measure Theory

I have an example from class notes that I do not understand and would appreciate some clarification. Particularly, I haven't found a direct explanation online or in my text with regards to the limits ...
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1answer
26 views

Example of a Lebesgue integrable function under certain conditions

I need to find a sequence $(f_k)$ of Lebesgue integrable functions such that $f_k \to 0$ almost everywhere but $\lim _{k\to \infty} \int |f_k| \ne 0$. Here is an example that I thought: $f_k(x) = ...
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0answers
49 views

Antiderivative $f(x)=\int_{a}^{x} g(x)$ is differentiable almost everywhere on (a,b)

An exercise in Royden & Fitzpatrick asks to show that if g is integrable on [a,b] and we define the antiderivative of g as: $f(x)=\int_{a}^{x} g(x)$ for all $x\in[a,b]$, then f is differentiable ...
1
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3answers
74 views

prove that $ \int_E f(x) dm \ge \delta$ whenever $m(E) \ge \epsilon$

Assume that $f: [0,1] \to [0, \infty) $ is a Lebesgue measurable function such that $ f(x)\gt 0 $ a.e x. Show that for every $ \epsilon \gt 0 $ there is $\delta \gt 0$ such that for every lebesgue ...
0
votes
1answer
37 views

If given conditions are satisfied, then prove that $f$ is absolutely continuous on any interval $[a,b]$

Assume that $ f: R \to R $ is a non-decreasing function with $ \int_R f' dm =1, $ $ \lim_{x \to-\infty} f(x) =0 $ , $ \lim_{x \to\infty}f(x)=1 $. Then Prove that $f$ is absolutely continuous on any ...
1
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1answer
37 views

$ \lVert {\bf f} \rVert_{p} \leqslant (m(E))^{1-1/p} \lVert {\bf f} \rVert_{\infty} $ for any $ f \in L_{\infty}(E) $

If $m(E)$ is finite and ${\bf f}\in L_\infty(E)$ then for any $p\geqslant 1$, $$ \lVert {\bf f} \rVert_{p} \leqslant (m(E))^{1-1/p} \lVert {\bf f} \rVert_{\infty} .$$ I tried to apply Hölder's ...
1
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3answers
36 views

Show that there is no Lebesgue integrable function $g$ satisfying $n\chi_{(0,\frac1n]}\leqslant g$ for all $n$.

Show that there is no Lebesgue integrable function $g$ satisfying $n\chi_{(0,\frac1n]}\leqslant g$ for all $n$, where $\chi$ is the characteristic function. I tried to reason by contradiction. ...
1
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0answers
25 views

Convergence in norm

If {$f_k $} is a sequence of Lebesgue integrable functions, then {$f_k$} is said to "converge in norm" to an integrable function $f$ if $\int | f_k - f | $ converges to zero . Can someone explain to ...