Tagged Questions

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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0
votes
1answer
74 views

Understanding this inner product

I want to find out under which conditions on $w$, we have that $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x)w(x) dx $$ a dot product?, where $f,g \in C([0,1],\mathbb{C})$ and $w \in ...
2
votes
1answer
124 views

Hardy-Littlewood maximal function weak type estimate

Show that if $f\in L^1(\mathbb{R}^d)$ and $E\subset \mathbb{R}^d$ has finite measure, then for any $0<q<1$, $$\int_E |f^{*}(x)|^q dx\leq C_q|E|^{1-q}||f||_{L^1(\mathbb{R}^d)}^{q}$$ where $C_q$ ...
3
votes
2answers
88 views

How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel-measurable and Lebesgue-integrable over $I:=(0,1)$. Further, let $\;\;\;\;\;\;\;\;\;\;g : I\to \mathbb{R}\;,\;\;\; \displaystyle x ...
5
votes
2answers
200 views

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

I'm asked to prove $$\displaystyle\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}\tag{$\ast$}$$ by integration of $e^{-x}\text{sin}(2xy)$ over an suitable measurable ...
5
votes
0answers
48 views

Prove that the lebesgue integrals $\lim_{n\to \infty} \int_{[0,1]^n} \frac{x_1^2+ \dots +x_n^2}{x_1+ \dots +x_n} \ d \mathrm{m}(\mathbf{x})=2/3$ [duplicate]

I found this problem while studying for an analysis exam and have been puzzled ever since. If anybody has any hints on this, I would greatly appreciate it. Prove that the lebesgue integrals ...
2
votes
1answer
66 views

Invariance of integral

Given the Lebesgue integral with the Lebesgue measure and the Borel-Sigma Algebra, I am supposed to figure out under which transformations $\int_{\mathbb{R}^2} f(x) dx$ the integral is ...
2
votes
1answer
155 views

Integration, Lebesgue and counting measure

Could you help me with the following exercise? Consider $X=Y=[0,1]$ with Lebesgue measure $m$ on $X$ and counting measure $\omega$ on $Y$. Let $f:X \times Y \rightarrow \mathbb{R}$ and $f(x,y)= ...
1
vote
1answer
126 views

Is the limit of an uniformly convergent sequence of integrable functions (with convergent integrands) integrable?

My question is the following: Consider a sequence of Lebeque integrable functions $f_{n}$ (over R) that converges uniformly. Assume furthermore that the integrands $\int f_{n}$ converge to some ...
5
votes
1answer
92 views

If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, is it true that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$.

If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, p,rove or disprove that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$. I think it is true. It is easy to see ...
2
votes
3answers
100 views

Give a proof/hint or counter example of the following “cool” real analysis question

Let $a_n$ be a sequence and $N\in\mathbb{N}$ be fixed. Assume that no more than $N$ of the $a_n$'s lie in any interval of lenght $1$. Show that for any $f\in L^1(\mathbb{R})$, we have ...
1
vote
1answer
86 views

Convergence in $L^1$ of defined step functions

Question: Let $f \in L^1(m)$. For k=1,2,... let f_k be the step function defined by $$ f_k(x)=k\int_{j/k}^{(j+1)/k} f(t)\,dt $$ $$ \text{for } \frac{j}{k}<x\leq\frac{j+1}{k}, \quad ...
1
vote
1answer
59 views

WTS $\int|f(t+s)-f(t)|dt\rightarrow ?$ as $s\rightarrow 0$

so in looking at this one qual... part a was this one, part b was show what the limit as $s\rightarrow\infty$. ($f\in L^1$) For both the answer is 0; I get that, but I'm trying to insure the proof ...
0
votes
1answer
58 views

Question about integrable functions and integrands

Suppose that $f \in L(X, \mathbb{X}, \mu)$ and that $\lambda(E)=\int_E f d\mu$ for all $E \in \mathbb{X}$. Show that $\lambda(E) \geq 0$ for all $ E \in \mathbb{X}$ if and only if $f(x) \geq 0$ for ...
1
vote
4answers
73 views

Question of Lebesgue integrals to show functions are the same \mu a.e

Let $f$ and $g \in L(X,\mathbb{X},\mu)$ be such that $\int f d \mu = \int g d\mu < \infty$ and $\int_E f d \mu = \int_E g d\mu$ for all $E \in \mathbb{X}$ Show that $f(x)=g(x) \mu$ a.e $x$. I ...
0
votes
1answer
59 views

Lebesgue Integral: calculation of limit

I need to calculate the following limit, using basic Lebesgue integral properties and lemmas: $$\underset{n\rightarrow\infty}{\lim}\underset{}{\int_{(a,\infty)}\frac{n}{1+n^{2}x^{2}}}.$$
2
votes
1answer
136 views

Notation used for integrals w.r.t. probability measures

I was staring at one of my questions at SE and realized that I do not really understand what I mean by $dP(\omega)$ when I write: $$EX = \int_{\Omega} X(\omega) \, dP (\omega)$$ where $X: \Omega \to ...
2
votes
1answer
58 views

Computation of $\lim_{n \rightarrow \infty} n(\int_{0}^{\infty} \frac{1}{1+x^4+x^n} \mathrm{d}m(x)-C)$

Let $m(x)$ be the Lebesgue measure. I want to show that there exists $C\in \mathbb R$ such that $$\lim_{n \rightarrow \infty} n\left(\int_{0}^{\infty} \frac{1}{1+x^4+x^n} \mathrm{d}m(x)-C\right)$$ ...
0
votes
1answer
44 views

Enumerable unions and Lebesgue measure; Result from Macbeath & Rogers

Let $f:(0,1] \rightarrow \mathbb{R}$ be measurable function and $T \subset SL(n, \mathbb{R})$ such that the set $T(1) = \{ \lambda A: A \in T, \lambda \in (0,1] \}$ is measurable. Then: $\int_{T(1)} ...
4
votes
1answer
82 views

Exercise of integration

I have to solve this: $$ \lim_{n\rightarrow\infty}\int_0^1\frac{e^{-nt}-(1-t)^n}{t}dt $$ (hint: $1-e^{nt}(1-t)^n=\int_0^t ne^{\tau n}\tau(1-\tau)^{n-1}d\tau$). I wrote that: $$ ...
1
vote
1answer
46 views

Compute $\lim_{n\rightarrow \infty} n\int_0^1(1+x)^n e^{-3nx} \mathrm{d} \mu (x)$

Compute $\lim_{n\rightarrow \infty} n\int_0^1(1+x)^n e^{-3nx} \mathrm{d} \mu (x)$ Well, I think I know how to solve this one. I have to use the Lebesgue dominated convergence theorem, so that the ...
0
votes
1answer
59 views

Show that $\int_0^\infty f du = \lim_{⁡n\to\infty}\int_0^n fdu$

Show that if $f$ is Lebesgue integrable on $[0,\infty)$ then $$\int_0^\infty f du = \lim_{⁡n\to\infty}\int_0^n fdu$$ I tried using Fatou's lemma but I failed to obtain the result. How to go ...
1
vote
1answer
67 views

Theory of “integration” on sets of measure zero (the measure zero sets are with respect to Lebesgue measure )

Suppose we have Lebesgue measurable set E. Let F be a subset of E with measure zero with respect to the Lebesgue measure on E. My question is, can we construct a "reasonable" integration theory on the ...
2
votes
1answer
95 views

Lebesgue measure and integral

i have a question about Lebesgue measure and integration. We have the follwoing situation: Let $(q_n)_n$ be a counting of the rational numbers $\mathbb{Q}$. Define $g:\mathbb{R}\rightarrow[0,\infty)$ ...
0
votes
1answer
98 views

Why any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset homeomorphic to $[0,1]$

as a continuation to my question here: Is cantor set homeomorphic to the unit interval? I can't see how can it is be true that any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset ...
2
votes
1answer
51 views

Prove that there exists $h \in L^1(\mu)$ such that $h \geq 0$ and $\int f \circ T = \int fh$

I am studying for the qualifiers and am trying to solve the following question: Let $(X, \mu, \mathscr{M})$ be a finite measure space. Suppose $T: X \to X$ is measurable and $\mu(T^{-1}(E))=0$ ...
2
votes
1answer
93 views

Question regarding Lebesgue Integrability in $\sigma$ -finite spaces

I'm taking a course in measure theory and we defined integrability in a $\sigma$ -finite space as follows: Suppose $\left(X,\mathcal{F},\mu\right)$ is a $\sigma$-finite measure space, a measurable ...
1
vote
1answer
74 views

Integral of the product of an bounded and any continous functions

If $u:[0,1]\rightarrow \mathbb{R}$ is bounded measurable function so that for all $v\in C[0,1]$ $$\int_0^1uvdx=0$$then show that u is zero almost everywhere on $[0,1].$ Thanks in advance for any ...
0
votes
1answer
40 views

Integral converges in $E^n$

How do I prove that $\displaystyle \int_{E^n}|\mathbf{x}|^{-|\mathbf{x}|}d\lambda(\mathbf{x})$, where $\lambda$ is lebesgue measure, converges? I was thinking of finding an upper bound function for ...
1
vote
1answer
81 views

Give a example in Lebesgue integral

Please help me to solve the following problem that is in the Lebesgue integral discussion Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that ...
6
votes
3answers
482 views

Prove that $\,f=0$ almost everywhere.

Let $f$ be a Lebesgue integrable function on $[0,1]$ such that, for any $0 \leq a < b \leq 1$, $$\biggl|\int^b_a f(x)\,dx\,\biggr| \leq (b-a)^2\,.$$ Prove that $f=0$ almost everywhere. I would ...
2
votes
2answers
158 views

Does uniform integrability plus convergence in measure imply convergence in $L^1$?

Does uniform integrability plus convergence in measure imply convergence in $L^1$? I know this holds on a probability space. Does it hold on a general measure space? I have tried googling. It ...
3
votes
4answers
90 views

$A \subset \mathbb{R} $ is measurable, prove that $-A=\{x : -x \in A\}$ is measurable.

$A \subset \mathbb{R} $ is measurable, prove that $-A=\{x : -x \in A\}$ is measurable. It is more than obvious that $-A$ is measurable, but I am sure that I am not supposed to say :"$-A$ is just $A$ ...
1
vote
1answer
59 views

A continuous $L^1$ function $f : \mathbb{R}\rightarrow \mathbb{R}$..

For a continuous function $f :\mathbb{R}\rightarrow \mathbb{R}$ satisfying $$\int_{\mathbb{R}}|f(x)|dx<\infty$$ and for some $\alpha >0$ let $d_f(\alpha)$ be tthe lebesgue measure of the set ...
1
vote
1answer
170 views

Example of a countably infinite set that has a positive volume

Definition: The volume of a bounded set $A\subset\mathbb R^n$ whose characteristic function $1_A$ is integrable over $\mathbb R^n$ is $\int_A 1_A$. I'm looking for an example of a countably infinite ...
3
votes
1answer
80 views

lebesgue measure/Measurable sets

Question is : Let $f,g$ be measurable real valued functions on $\mathbb{R}$ such that : $$\int_{-\infty}^{\infty} (f(x)^2+g(x)^2)dx=2\int_{-\infty}^{\infty} f(x)g(x)dx$$ Let $E=\{x\in \mathbb{R} ...
2
votes
1answer
54 views

Two questions of the measure theory

(1) Suppose the function $g(x) = \sqrt{|f(x)|}$ is measurable on $[0,1]$ with respect to classical Lebesgue measure. Does it follow that the function $f(x)$ is measurable on $[0,1]$? (2) And ...
0
votes
1answer
37 views

Sequences for which $\lim_n \sin(a_nx)$ exists on a set of positive measure

Let $E \subset \mathbb R$ be a set of finite (Lebesgue) measure and let $\alpha_n \to +\infty$. Then prove $$ \lim_{n} \int_E 2\sin^2{(\alpha_nt)} dt = \mathscr{L}^{1}(E) $$ Using the ...
1
vote
0answers
43 views

Product of a singular and continuous measures

Suppose we have a function $f(x,y)\geq 0$ and we integrate it with respect to a measure $\mu=\mu_{\mathrm{a.c.}}+\mu_{\mathrm{disc}}+\mu_{\mathrm{s.c}}$. Is it true that \begin{equation} \iint ...
4
votes
1answer
50 views

Problem involving Lebesgue Integral

Suppose $f\in L^1(\mathbb{R}^2)$ is s.t. $||f||_{L^1}=1$. I want to prove that $\exists x\in\mathbb{R}^2$ with $|x|\leq 1$ AND $\int_{\mathbb{R}^2}f(y)|x-y|^{-1}dy<100$. I tried taking a double ...
1
vote
1answer
173 views

Understanding Stokes' theorem

Stokes' theorem( here I am only talking about the special $\mathbb{R}^3$ case) contains a line integral $\int_{\partial S} \langle f, \tau \rangle ds$. (Actually, I would be confident if somebody ...
0
votes
1answer
98 views

Prove that, $f*\phi_{\delta}(x)\overset{(\delta\rightarrow 0)}\longrightarrow f(x)\ ,\forall x\in\mathbb R$ (where * is the convolution)

If $\phi:\mathbb R\rightarrow\mathbb R$ with, $\int_{\mathbb R}\phi(x)\ dx=1$ and $\phi_{\delta}$ is defined by, $\phi_{\delta}(x)=\frac{\phi(\frac{x}{\delta})}{\delta}$. Prove that, for every ...
2
votes
2answers
103 views

Is $C[a,b]$ a closed linear subspace of $L^{p}(a,b)$

I am not sure about the last step of my proof: $(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). ...
0
votes
2answers
76 views

A question about Lebesgue integrable function on a measurable set

Show that if $f$ is a Lebesgue integrable function on a measureable set $E$ of finite measure and $E_n=\{x \in E : \lvert f(x)\rvert \geq n\}$, then $\lim_{n \to \infty} (n\cdot m(E_n)) = 0$, where ...
3
votes
2answers
66 views

Exchanging the limit and Lebesgue integral for every set in the measure space

Let $(X,\mathcal{M},\mu)$ be a measure space and let a sequence of $\mathcal{M}$-measurable functions $\{f\}_k$, where $f_k: X\rightarrow [0,\infty]$. Assume that $f_k\rightarrow f$ pointwise and ...
2
votes
1answer
56 views

Differentiability of function defined as integral form

Let $H(t)=\int_{\Bbb R}|f(x)+tg(x)|^p\mathrm dx$ and $f,g\in L^p(\Bbb R)$. Then, how to prove that $H$ is differentiable and find its derivative? I think it's impossible to find it by ...
0
votes
0answers
20 views

Limit of $L^r$ norm in lebesgue measure theory [duplicate]

Let $f\in L^r$ for some $r>0$ and $\mu (X)=1$. Then, prove that $\lim_{p\to 0}||f||_p=\exp(\int \log|f|d\mu)$. This is from chapter $L^p$ spaces, but I don't have any idea. How to make $\log$? ...
1
vote
2answers
158 views

$f^2$ integrable and $f$ is not

I'm trying to find an example of a function that is not Lebesgue integrable but $f^2$ is integrable. The problem I am trying to solve includes the converse for which i gave: $\frac{1}{\sqrt{x}}$ or ...
2
votes
0answers
50 views

Integrability and product measure

Let $X$ and $Y$ be subsets of $\mathbb{R}$, and let $\mu$ be a measure on $X$ and $\nu$ a measure on $Y$. Let $f : X \times Y \rightarrow \mathbb{R}$ be $\mu$-summable and $\nu$-summable, i.e. ...
1
vote
2answers
48 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
0
votes
1answer
55 views

Sufficient and necessary condition for Lebesgue integrability of a random variable

Could anyone give me a hand with the following problem? Let $f$ be a random variable over a probability space $(\Omega,A,\mathbb P)$. Show that $f$ is integrable $\iff $ ...