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
19 views

Integral of the characteristic function.

I know $\chi$ is the characteristic function. Why does $\int \chi_I= L(I)$? where L(I) is the length of I.
0
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1answer
20 views

Problem with Lebesgue-integral of measurable set and Lebesgue-integrable function

I am trying to show, that Let $\Omega \subset \mathbb{R}^n$ be measurable and $f\colon \Omega \rightarrow [0,\infty)$ Lebesgue-integrable. Show that: $$\int_\Omega f(x)dx=\int_0^\infty ...
3
votes
1answer
48 views

Eigenvalues of an integral operator

The following operator is defined on $L_2(0,1)$: $$Kf(t)=\int_0^1|s-t|f(s)ds$$ I am wondering how I can calculate the eigenvalues and eigenfunctions of such an operator. I start with ...
1
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2answers
19 views

Show that $f_n\to f$ in the norm $L^1(\mathbb{R})$ for $f\in L^1(\mathbb{R})$.

Let $f\in L^1(\mathbb{R})$. Define $$f_n(x)=\begin{cases} f(x) & \text{if }|x|\leq n\\0 & \text{otherwise}\end{cases}.$$ Show that $f_n\to f$ i.n. This seems really obvious, so I'm not sure ...
1
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1answer
33 views

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions.

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions. I'm not sure where to even start with this. The question doesn't ...
3
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1answer
84 views

Let $f$ be integrable over $\mathbb{R}$. Show that the following four assertions are equivalent:

Let $f$ be integrable over $\mathbb{R}$. Show that the following four assertions are equivalent: $f = 0$ a.e. on $\mathbb{R}$. $\int_{\mathbb{R}}fg=0$ for every bounded measurable ...
0
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1answer
28 views

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$ To my knowledge I need to use Fubini's theorem. But this doesn't work because the integration would ...
0
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2answers
40 views

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable.

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable. I've shown that it isn't Reimann integrable, but I'm stuck on showing it is Lebesgue integrable. Any help would be ...
0
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1answer
32 views

Prove a function to be analytic by dominated convergence theorem

Given $f \in L^1$, prove that $$F(z) = \frac{1}{2\pi i} \int^\infty_{-\infty} \frac{f(t)}{t-z}\,dt$$ is an analytic function and $$F'(z)=\frac{1}{2\pi i} \int^\infty_{-\infty} ...
1
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1answer
18 views

A bounded family of functions in $L^p[E]$, where E is a measurable set, is uniformly integrable.

A corollary in Royden & Fitzpatrick's Real Analysis (chapter 7 section 2) reads: Let $E$ a measurable set, and $1<p<\infty$. Suppose $F$ is a family of functions in $L^p(E)$ that is bounded ...
2
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1answer
52 views

Lebesgue monotone convergence theorem

I have a doubt regarding the Lebesgue monotone convergence theorem. The version that I know is the following from Wikipedia, requiring, in particular, $\{f_k(x)\}$ monotone increasing and ...
1
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1answer
18 views

Prove sum is commutative with measure theory result

Given a measured space $(X,\mathcal{A},\mu)$, consider the following function $f:X\rightarrow \mathbb{R}^+$ defined by $$ f(x)=\sum_{n=0}^\infty f_n(x), $$ where $f_n(x)$ are positive measurable ...
1
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0answers
38 views

$L^p$ space and continuous injection

Let $1\leq p < r < q \leq \infty$ and $E\in \mathbb{R}$. Define $$A = L^p(E) + L^q(E) = \{f=g+h:g\in L^p(E), h\in L^q(E) \}$$ and $$\|f\|_A = \inf_{f=g+h} \|g\|_p+\|h\|_q$$ where the infimum ...
1
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1answer
57 views

Prove $\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$

I try to prove the following statement: $$\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$$ There is also a clue: $$ ...
0
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1answer
36 views

Is it always true that the Lebesgue integral of a continuous function is equal to the Riemann integral (even if they are both unbounded)?

Let's assume that $f\colon\mathbb R\to\mathbb R$ is continuous and hence Lebesgue measurable. Then, the Lebesgue integral $\int_{(0,\infty)}f(x)\,d\lambda(x)$ makes sense (but of course can be equal ...
2
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1answer
53 views

For every open subset of $\mathbb{R}^n$ there is a countable non-overlapping union of closed squares

I am struggling with a definition which should lead to the Lebesgue outer measure. Theorem: Every open subset of $\mathbb{R}^n$ is a countable disjoint (with the meaning non-overlapping or the ...
1
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0answers
25 views

Prove convergence in $L^1$

Let $(X, \mathscr{A}, \mu)$ be a finite measure space. Let $f_n \in L^1$. Assume $f_n \rightarrow f$ a.e. and there exist $p > 1$ and $c > 0$ such that $$||f_n||_p < c$$ for all $n$. I want ...
2
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0answers
26 views

How to use dominated convergence on $\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x) $?

I find it hard to find an appropriate dominating function for the integral $$I:=\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x), \ t > 0 $$ ...
1
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1answer
43 views

Integration of nonnegative funtion, Folland Real Analysis

Suppose $f$ is a nonnegative measurable function on a measure space $(X,M,\mu)$ satisfying $\int f d\mu < \infty$. Show that for every $\epsilon > 0 $ there exists a $\delta > 0 $ such that ...
0
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3answers
50 views

Show that $ \int \liminf f_n \leq \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n$

Let $g$ be a non-negative integrable function over $E$ and suppose $\{f_n\}$ is a sequence of measurable functions on $E$ such that for each $n$, $|f_n| \leq g$ a.e. on $E$. Show that $$ \int ...
1
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1answer
46 views

Why does the Lebesgue integral equal the Riemann integral here?

We have $$ \mathbf{E}[g(X)] \stackrel{\text{df}}{=} \int_{\Omega} g(X) ~ \mathrm{d}{P} = \int_{\Bbb{R}^{k}} g ~ \mathrm{d}{P_{X}}, $$ where $ X: \Omega ...
1
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1answer
39 views

Use MCT on $(2x-4)^{-1/2}$

Use Monotone Convergence Theorem to prove that $$f(x) = [(2x-4)^{-1/2}]1_{(2,4]} + 01_{\{2\}}$$ is Lebesgue integrable on $[2,4]$. According to notes from a different class So I just replace $x$ ...
0
votes
2answers
15 views

Proving Lebesgue integral statements

Let $(X, \Sigma, \mu )$ be a measure space. If $f: X \rightarrow \bar{\mathbb R}$ is measurable and $\int |f| \, d \mu < \infty$, then: for any $a>0$, the set $\{x \in X : |f(x)|>a \}$ has ...
0
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1answer
32 views

Evaluate $\int_{[0,6]} \sin \frac{\lfloor x \rfloor \pi}{4} d\lambda$ / Notation check

Given a measure space $(S, \Sigma, \mu) = ([0,6], \mathscr B([0,6]), \lambda)$, evaluate $$\int_{[0,6]} \sin \frac{\lfloor x \rfloor \pi}{4} \lambda(dx)$$ $$\int_{[0,6]} \sin \frac{\lfloor x ...
1
vote
1answer
33 views

Find $\int _{[0,1]} f \, \, d \lambda$

Let $e_k(x)$ be the $k$-th digit after the decimal point $x$ written as a decimal. Let $f(x) = e_2(x)$ (so if $f(0.521)=2$) Find $$\int _{[0,1]} f \, \, d \lambda$$ I have no idea how to do this. ...
1
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0answers
56 views

If $\int f(x)g(x) dx = 0$, then $f = 0$ almost everywhere in $\Omega$

How do I show, that $\text{Let $\Omega \subset \mathbb{R}^n$ be open. Satisfies } f\in\mathcal{L}^1(\Omega) \text{ following propertie }$ $$\int_{\Omega} f(x)g(x) dx = 0 \text{ for all } ...
0
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0answers
24 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 ...
3
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0answers
19 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 ...
1
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1answer
30 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 ...
1
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2answers
64 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
votes
0answers
53 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 ...
0
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1answer
18 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
votes
0answers
17 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 ...
0
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1answer
27 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 ...
0
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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 ...
0
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0answers
26 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
55 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 ...
1
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1answer
43 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 ...
1
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1answer
110 views

when composition of continuous and Lebesgue integrable function Lebesgue integrable

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
54 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 ...
1
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0answers
16 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
votes
1answer
46 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 ...
0
votes
1answer
28 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 $f,g \in L^1(\mathbb{R})$, ...
2
votes
1answer
49 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 ...
1
vote
0answers
12 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
votes
0answers
29 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 ...
1
vote
1answer
49 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
votes
1answer
24 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
votes
0answers
39 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 ...
1
vote
2answers
57 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 ...