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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40 views

Regularity of a measure $n(E) = \int_{E} f(x) dx$

I would like to show that the (positive) measure $n$ on $\Bbb R \setminus \{0\}$ defined by $n(E) = \displaystyle \int_E \frac{dx}{|x|}$ is outer regular ($dx$ being the usual Lebesgue measure). ...
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0answers
15 views

Is the charateristic function $\chi _{\Omega }$ in the Sobolev space $W^{1,2}_{0}(\Omega)$?

Given $\Omega$ is a bounded, $C^1$ domain in $\mathbb{R}^n$. $\chi _{\Omega }(x)$ is the characteristic function of $\Omega$. I have done the followings: We can get $\chi _{\Omega }(x) \in ...
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1answer
15 views

translation invariant of integral on $\mathbb{R}$

Problem 1: Let $f\in L^1(\mathbb{R})$, show $\int_\mathbb{R}f(t)dt=\int_{\mathbb{R}}f(x+t)dt,\forall x\in (-\infty, \infty)$. Problem 2: Let $f\in L^1(\mathbb{R})$,show $\displaystyle ...
2
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0answers
48 views

Understanding principal value integral

I'm reading the original article on distance covariance (link), and throughout the article the author uses the following lemma: Can someone please explain what he actually means by "principal value ...
1
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2answers
63 views

If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.

I'm learning about measure theory, specifically the Lebesgue intregal of nonnegative functions, and need help with the following problem. Let $f:\mathbb{R}\to[0,\infty)$ be measurable and $f\in ...
2
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2answers
60 views

Calculate limit with integral

Hi I have a problem with following limit: $$\lim_{x\rightarrow\infty}e^{-x}\int_{0}^{x}\int_{0}^{x}\frac{e^u-e^v} {u-v}\ \mathrm du\ \mathrm dv$$ as a hint i got that i should use de l'Hospital. So: ...
2
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1answer
26 views

Why is the fact true that if $E$ has measure zero, then $E_a$ has measure zero?

I am proving the following theorem: For a subset $E$ of $\mathbb{R}^n$, if $E$ is measurable, then $E_a$ is measurable and $|E_a| = a|E|$. where $E_a := \{(\mathbf{x}, y):\mathbf{x}\in E, 0\le ...
-1
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1answer
36 views

Interesting relationship between cardinality and Lebesgue outer measure

If two sets $A$ and $B$ defined on bounded intervals have the same cardinality and $ A \bigcap B $ is non empty and the Lebesgue outer measure of A is greater than zero. Is it then true that the ...
2
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2answers
43 views

Let $f:\mathbb{R}\to[0,\infty)$ measurable and $f\in L^1$. Show that $\mu(E)<\delta \implies \int_E f < \varepsilon$.

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help to understand the solution to the following problem: Let $f:\mathbb{R}\to[0,\infty)$ ...
5
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1answer
50 views

Reciprocal for Fubini-Tonelli's theorem

By Fubini-Tonelli's theorem, we know that if $E\in \mathbb{R^{n+m}}$ and $f: \mathbb{R^{n+m}}\to \mathbb{R_{>0}}$ are measurable and $f$ integrable, then the sections $E_x=\{y\in \mathbb{R^m}: ...
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2answers
24 views

Show that $\int\vert \int f(y)g(x-y)\,dy \rvert \,dx \le \iint \vert f(y)g(x-y)\rvert\, dy\, dx$

I'm reading a theorem as following about the convolution, but didn't understand the first step: Why is this inequality true? i.e. why is $$\int\left| \int f(y)g(x-y)\,dy \right| \,dx \le \it \vert ...
1
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1answer
24 views

An integrally defined borel measurable function

Let $f:\mathbb{R} \rightarrow (0, \infty)$ be a Borel measurable function and let $E$ be a Borel measurable subset of $\mathbb{R}$ such that $\lambda (E) > 0$. Define $F(t) = \int_{E}f(t + x) d ...
1
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0answers
28 views

Show that if $f^{-1}(A)$ is Borel measurable, then $f$ is also Borel measurable

Let $A$ be a Borel measurable set. I want to show that if $f^{-1}(A)$ is Borel measurable, then $f$ is also Borel measurable. I know the other direction is true, but I'm not sure if my claim is ...
0
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2answers
111 views

Shortest Proof of Lebesgue Dominated Convergence Theroem ( 5 lines) without using Fatou's lemma

If {$G_n$} is a sequence of bounded measurable functions and $ | G_n | \le M $ where M is a positive real number $\lim\limits_{n\mapsto \infty} G_n =F$ on a bounded measurable set E , $\epsilon> ...
2
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1answer
52 views

Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality $\int_{0}^{\infty}e^{-tx}\,dx=\frac{1}{t}$

I'm learning about measure theory, specifically Lebesgue integral, and need help with the following problem: Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality ...
4
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2answers
63 views

Prove that $F(x,y)=f(x-y)$ is Borel measurable

Suppose $A$ is a subset of $\Bbb R$, let $s(A)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in A\}$. I already showed: If $A\in \Bbb B$ (Borel measurable set), then $s(A)\in \Bbb B \times \Bbb B$. I want ...
2
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1answer
67 views

If $f:\mathbb{R}\to[0, \infty)$ (uniformly) continuous and $f \in L^1$, then $\lim_{x\to\pm\infty}f(x)=0$?

I'm learning about measure theory and need help with the following questions: True or False (justify): $(1)$ If $f:\mathbb{R}\to[0, \infty)$ measurable and $f \in L^1$, then ...
1
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1answer
17 views

Compute $\int_{B(0,2)} (xy^2+y^2z)d\mu(x,y,z)$ where $\mu$ is defined by an another integral

I have a measure $\mu$ defined as follow: for every a measurable set $E \subset \mathbb{R}^3$ $$\mu(E)=\int_{E \cap B(0, 1)} \sqrt{x^2+y^2+z^2}d\lambda_3(x,y,z)$$ where $\lambda_3$ is the Lebesgue ...
6
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1answer
73 views

Prove series converge for almost every $x$

Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in ...
1
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2answers
60 views

Proving that $\left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3$, given $1\leq a<b$

If $1\leq a<b$, then $$ \left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3.$$ Proceeding by integration by parts; let $u(x)=\sin(x)$ and $dv(x)=1/x$, then $u'=\cos(x)$ & $v(x)=\log(x)$. We ...
2
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2answers
57 views

Part of proof to show Lebesgue-lebesgue measurable

I want to prove the following: Suppose $E$ is a subset of $\Bbb R$, let $\gamma(E)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in E\}$. If $E\in \Bbb B$ (Borel/Lebesgue measurable set), show that ...
3
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1answer
19 views

Approximate an integrable function using a simple function (Proving existance)

Let $f \in L^1(\mathbb{R})$, and let $\epsilon > 0$. Show that exists simple function $g=\sum_{k=1}^{n}c_k 1_{A_k}$, such that, $$\int_\mathbb{R} |f(x)-g(x)|dx \leq \epsilon$$,and such that $n \in ...
1
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2answers
39 views

Helping understand line integral $\int_{K,+}{(x+y)}dx+(y-x)dy$

I have a huge problem with understanding line integrals and would be much obliged for your help! We have: $$\int_{K,+}{(x+y)}dx+(y-x)dy$$ and the following parameterization: ...
0
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1answer
34 views

Lebesgue integrals and polinomial functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non zero polinomial function. Prove that $f\notin\mathbb{L}(\mathbb{R})$.In other words is not Lebesgue integrable.
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0answers
10 views

If $| \int_A fdλ| ≤ λ(A)$ for $A \subset [-1,1]$. then range of f contained in [-1,1]

Let f : [−1, 1] → R be a continuous function. Let λ be the Lebesgue measure on [−1, 1]. Suppose $| \int_A fdλ| ≤ λ(A)$ for all measurable sets A ⊆ [−1, 1]. I want to show that the range of f is ...
0
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1answer
28 views

Inequality of Lebesgue integrals

Let $f,g\in\mathbb{L}(E)$. Suppose that $f\leq g$ and $A:=${$x\in E| f(x)<g(x)$}. Prove that $\int_{E}f<\int_{E}g$ if and only if $A$ has positive measure.
2
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1answer
23 views

Continuous functions are locally integrable?

If $K\subset\mathbb{R}$ is compact and $f:K\rightarrow\mathbb{R}$ continuous then $f\in\mathbb{L}(K)$. In other words $f$ is integrable in $K$. So far i know that since $f$ is continuous then $f(K)$ ...
2
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0answers
78 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
2
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1answer
34 views

Application of Dominated Convergence Theorem help finding a Dominating function

$$\lim_{n\to\infty}\int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)}$$ I wish to use the Lebesgue Dominated Convergence theorem to solve this, but I'm having trouble finding a dominating function, $g(x)$. ...
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2answers
78 views

Explicit Integrals and LimInf/LimSup

(a) Show that $f(t):=\int_0^\infty e^{-tx}\frac{sin \space x}{x}dx$ exists for $t>0$ and defines a differentiable function $f$. Calculate $f'(t)$ for $t>0$ and evaluate it explicitly. (b) Prove ...
0
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1answer
17 views

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

I discovered last night that I have an error in my proof to the following problem and I need help fixing it (or need a new solution) $$ \text{Suppose that} \int_{0}^\infty x^{-2}|f|^5 dx < \infty. ...
0
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1answer
14 views

Inequality regarding measure of function and integral of function

Let $(X,\Sigma,\mu)$ be a measure space. Let $f$ be a measurable function and $t > 0, t\in \mathbb{R}.$. Denote: $$C_f(t) = \mu \{x \in \Omega : |f(x)| \geq t \}.$$ In the first part of ...
1
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1answer
28 views

Justifying the differentiation property of the Fourier transform

Let the Fourier transform of $f\in L^1(\Bbb R)$, denoted by $\mathcal{F}f$, be defined as $$ \mathcal{F}f(y) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ixy} f(x)\,dx.$$ An oft-quoted result ...
0
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1answer
27 views

small shift in domain of simple function

Suppose $s:[0,1]\to\mathbb R$ is a simple function with $s(0)=s(1)$. Let $S:\mathbb R \to \mathbb R$ be $S(x)=s(x-\lfloor x\rfloor)$, the function that repeats $s$ on each interval $[n,n+1]$. I'm ...
3
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1answer
75 views

Part of proof of the set of continuous integrable functions is dense in $L^1(\Bbb R)$

I want to prove: If $g$ belongs to $L(\Bbb R, \Bbb B, \lambda)$ and $\epsilon\gt 0$, then there exists a continuous function $f$ such that $\Vert g-f\Vert_1=\int \lvert g-f\rvert \,\text{d}\lambda \lt ...
2
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2answers
57 views

If a measurable function $f$ has zero integral over every measurable set *of finite measure*, then $f=0$ a.e.?

Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a regular measure on $X$. Suppose that $g : X \to \Bbb C$ belongs to $L^{\infty}(X)$. My question is : Is it sufficient to assume ...
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0answers
12 views

Convergence of two Lebesgue-Stieltjes integrals

I have I have a collection of bounded variation and right-continuous functions, $(F_n)_{n \in \mathbb{N}}$, and another bounded variation and right-continuous function, $G$, which satisfy $$\sup_x ...
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0answers
13 views

Is it a change of variable?

Hi everyone: In a book I am reading, they make a sort of "substitution" like this: let $B(0,R)$ be a ball in $\mathbb{R}^{N}$ $(N\geq2)$ and $f$ a locally integrable function. Let $\mu$ be a finite ...
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0answers
37 views

Approximating integrable functions

I wish to prove the following If $f$ is integrable and $f:\mathbb{R}\to\mathbb{R}$ then $$\lim_{t\to0}\int_{-\infty}^{\infty}|f(x)-f(x+t)| = 0$$ I have in my notes that if there exists a $g(x) \in ...
2
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4answers
173 views

Show $\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$

It's claimed that $$\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$$ by first expanding $\frac{\log(1-x)}{x}$ into a power series and then doing term-by-term integration. I want to justify this by ...
1
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1answer
97 views

Part of proof of term-by-term integration

I want to prove the theorem of term-by-term integration for lebesgue integrable functions (denoted as $L^1$ functions): Suppose $(g_n)$ is a sequence of $L^1$ functions over a measure space $(X,\sigma ...
0
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0answers
15 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
3
votes
1answer
46 views

Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Then $\int_{(0, 1)}\sum_{n=1}^{\infty}f_n \neq \sum_{n=1}^{\infty}\int_{(0, 1)}f_n.$

I'm learning about measure theory, specifically Lebesgue integration, and need help to understand the solution to the following problem: Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Show that ...
1
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2answers
61 views

Prove the monotone convergence theorem for sequences of Lebesgue-integrable functions

I'm trying to prove the monotone convergence theorem for $L^1$ functions: Suppose $(f_n)$ is a sequence of $L^1$-functions (i.e Lebesgue-integrable functions) over a measure space $(X,\sigma (X), ...
0
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0answers
25 views

conditions on integrable function with counting measure

Let $P(N)$ be the power set of $N$ and $u$ be the counting measure on $N$. (a) Prove or disprove the measure space $(N, P(N), u)$ is complete? (b) Given function $g: N\rightarrow R$. Show that $g$ ...
2
votes
1answer
28 views

Integration obeys countable subadditivity?

Does Lebesgue integration have the property of countable subadditivity: 'if $f$ is integrable on $E$ and $E = \bigcup_{i=1}^{\infty} E_n$ then $\int_E f \le \sum_{i=1}^{\infty} \int_{E_n} f$'? You ...
0
votes
0answers
61 views

Why does this integral not contradict Fubini's Theorem?

I have the integral: $$\int^{1}_{0}\int^{\infty}_{1} (e^{-xy}-2e^{-2xy}) \,\text{d}y~\text{d}x$$, and I know that the order of integration cannot be interchanged, but why does this not contradict ...
1
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0answers
32 views

When $F(t)=\int_0^tf(s)ds$ is differentiable everywhere?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function that is continuous almost everywhere. 1) Is the function $F(t)=\int_0^tf(s)ds$ differentiable everywhere ? 2) What is the "weakest" condition on $f$ ...
1
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1answer
25 views

Given $\lim\limits_{x\to\infty} f(x) = r$, show $\lim\limits_k\int_{[0,a]}f(kx) = ar$

Show $\lim\limits_k\int_{[0,a]}f(kx) = ar$ where $f:[0,\infty) \to \mathbb{R}$, bounded, Lebesgue measurable, and $\lim\limits_{x\to\infty} f(x) = r$. $$ \int_{[0,a]}f(kx) = \int \chi_{[0,a]}(x)f(kx) ...
0
votes
1answer
27 views

Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$ \int f d\mu $$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...