Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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42 views

$\sigma$-algebra of $\mathbb{R}$ generated by $\mathcal{P}(\mathbb{N})$

What is the $\sigma$-algebra of $\mathbb{R}$ generated by $\mathcal{P}(\mathbb{N})$? I thought it is $$\Sigma = \{\emptyset, \mathbb{N}, \mathcal{P}(\mathbb{N}), \mathbb{R}, \mathbb{R}-\mathbb{N}, ...
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0answers
37 views

Generating structure of Borel field

On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the author wrote: ...and there are Borel sets that cannot be arrived at from the intervals by any finite sequence of set-theoretic ...
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0answers
24 views

Example of a bounded simple process $A_t$ that changes value only once s.t. $\int_0^t A_s dB_s$ doesn't have normal distribution? [on hold]

As the title of the question suggests, what is an example of a bounded simple process $A_t$ that changes value only once such that$$\int_0^t A_s\,dB_s$$does not have a normal distribution?
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1answer
29 views

convolution of probability measures

What do we mean by convolution of measures? With example What is the difference between convolution of measures and convolution of functions? What is probability measure? Give an example of ...
2
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1answer
37 views

Measurability of marginal distributions of a random measurable function

For a probability space $(\Omega, \mathcal F, \mathsf P)$, let $X \colon \Omega \times [0,1] \to \mathbf R \colon (\omega, t) \mapsto X(\omega,t)$ be a random Borel function (i.e. an $(\mathcal ...
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1answer
35 views

Properties of decreasing sequence of Lebesgue measurable sets.

I'm trying to prove a property of Lebesgue measure sets that says: If the $A_{k}$'s are measurable and $A_{1} \supset A_{2} \supset A_{3} \supset \ldots,$ and if $\lambda (A_{1}) < \infty, $ then ...
2
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1answer
27 views

Help with a sigma-algebra problem with random variables (show $\sigma(X_S)\subseteq \sigma(X_T)$ if $S\subseteq T$)

My problem is as follows: Let $X_S$ and $X_T$ be two stochastic processes where $S,T$ are index sets. Let $\sigma(X_S)$ and $\sigma(X_T)$ denote the sigma-algebra generated by $X_S$ and $X_T$. ...
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65 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
2
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0answers
40 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
2
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1answer
42 views

Lebsegue measure of $\{ 0<x \leq 1: x \sin \left(\frac{\pi}{2x}\right) \geq 0 \}$

Find the Lebsegue measure of the set $A= \left\{ 0<x \leq 1: x \sin \left(\frac{\pi}{2x}\right) \geq 0 \right\}$. The answer given is $1 - \ln \sqrt{2}$. My thought: I only know that Lebsegue ...
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0answers
15 views

What is the difference between a “Borel probability measure” and a “singular Borel probability measure”? [closed]

What is the difference between a "Borel probability measure" and a "singular Borel probability measure"? When a probability meausure is said singular? Thanks in advance.
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0answers
22 views

Explicit constructions of Haar measures?

I know how to build the Haar measure somewhat explicitely on Lie groups (via differential forms) and profinite groups (by using the lemma that open subsets of a profinite group are unions of cosets of ...
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0answers
32 views

Basic Set-Theoretic Properties from Halmos

I've been backtracking lately to make sure that I have a solid set-theoretic background before taking measure theory this fall. Here's a few facts I've come across today, and my attempted proofs. Let ...
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0answers
34 views

Premeasure on $\mathcal{A}$ and $\mu^{*}$ proof

This proposition comes from Real Analysis by Folland: Some background information: (1.10) Let $\epsilon\subset P(X)$ and $p:\epsilon\rightarrow [0,\infty]$ be such that $\emptyset\in\epsilon$, ...
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0answers
34 views

Countable $\sigma$-algebra [duplicate]

Let $\Sigma$ be a countable $\sigma$-algebra. Show that there is a sequence $A_1,A_2,...$ of disjoint elements of $\Sigma$ such that every $B$ in $\Sigma$ is a countable union of elements in ...
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1answer
29 views

Constructing dependent sequences of random variables

It is easy, given some random variable $X \colon \Omega \to \mathbb{R}$ on a probability space $(\Omega, \mathbb{P})$, to construct an i.i.d. sequence $X_1, X_2, \ldots$ distributed as the law of $X$. ...
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1answer
51 views

Easy proof for existence of Lebesgue-premeasure

In the lecture on measure theory I attended last semester, we had a sort of complicated technical proof for the existence of the Lebesgue-premeasure. However, I can't see why this easier argument does ...
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0answers
46 views

$f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. [duplicate]

Problem: Suppose $f: \Bbb R \rightarrow \Bbb R$ is absolutely continuous on every interval $[a,b]$, and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. ...
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0answers
18 views

Relation between Borel sigma algebra on $\mathbb{R} $ and Borel sigma algebra on (n,n+1] [closed]

If A is a subset of (n,n+1] , n= natural number; then A belongs to borel sigma algebra on R , iff A belongs to borel sigma algebra on (n,n+1] interval . I need to prove the above statement . any ...
2
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1answer
32 views

Show that is a probability space

Let $ \Omega:= \{(x,y) \in \mathbb{R^2}:0<x,y \leq 1 \}$, let $\mathcal{F}$ be the collection of sets of $\Omega$ such that $$ \mathcal{F}:= \{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \}$$ ...
2
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2answers
29 views

Let $f:\Omega\to(0,+\infty)$ and $\ln(x)$ be $\mu$-integrable

Show that $\displaystyle \lim\limits_{p\to 0^+} ||f||_p = \exp(\int\ln(f)\,d\mu)$. In case it comes to be helpful. So far I've shown that $\displaystyle\lim\limits_{p\to ...
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1answer
27 views

About sets and algebras.

Let $\mathcal{F}$ be a collection of subsets os some nonempty set $\Omega$. Suppose that $\Omega \in \mathcal{F}$ and that $\mathcal{F}$ is closed under the formation of complements and finite ...
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22 views

Is the function $\,f(x,y)=y\,$ Lebesgue\Borel measurable? [closed]

Given function $\,f: \mathbb R^2\to \mathbb R ,\,$ establish whether $\,f(x,y)=y\,$ is Lebesgue and Borel measurable, and show that for every set $\,A\subseteq \mathbb R^2\, $ of type $\, ...
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0answers
22 views

pointwise converges and integrals

Let $(N,P(N),\mu)$ be a measure space such that $\mu(A)=\sum_{n\in A}{1\over n^2} $ a. Let $ f_n = n^2 * 1_{\{n\}} $. Does the sequence converges pointwise? b. Find all functions $ f:N \rightarrow R ...
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1answer
28 views

Borel measure induced by the Cantor function?

In an example to measure being mutally singular, the book has an example I do not understand. First the book has the definition: Mutually Singular Measure Let $(\Omega,\mathcal{A})$ be a ...
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1answer
41 views

A problem from Real Analysis of Folland

I got stuck on this problem. For the first statement, I tried to use $\epsilon -\delta$ condition, but still couldn't come to conclusion. So can anyone please help me solve this or give me some clue ...
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1answer
33 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
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2answers
60 views

If $\mu(|f_n|^p)$ is bounded and $f_n\to f$ in measure then $f_n\to f$ in $L^1$

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real measurable functions s.t., (a) The sequence $\displaystyle(\int |f_n|^p\ \mathsf d\mu)_{n\in\Bbb{N}}$ is bounded. (b) The sequence ...
2
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2answers
63 views

$\mu(A \cap I) \le a \mu(I)$ implies $\mu(A) = 0$?

Let $\mu$ be lebesgue measure on $\mathbb{R}$, $0<a<1$. If $\mu(A \cap I) \le a \mu(I)$ holds for any interval $I$, can I say $\mu(A)=0$? I tried to construct a counterexample by considering ...
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2answers
58 views

Show that a function defined by an integral is differentiable

Define $$g(a)=\int_{0}^{\infty}\frac{\sin(ax)}{x}e^{-x}dx,\ \ \ \ \ \ a\in\mathbb{R}$$ a) Show that $g(a)$ is differentiable and compute $g'(a)$. b) Use this to compute $g(a)$. I have tried various ...
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1answer
26 views

Let $(X,\mathcal{F},\mu)$ be a measure space and let $g\in L^1((X,\mathcal{F},\mu))$.

Let $\phi:[0,1]\to\mathbb{R}$ defined by $$\displaystyle \phi(t)=\int_X \frac{t^3g}{1+t^2g^2}\ \mathsf d\mu$$ Show that $\operatorname{Im}(\phi)\subset\mathbb{R}$ and that $\phi$ is continuous. ...
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0answers
14 views

Query about estimating an integral in Heat Equation

While studying the Heat Equation (P-309) from the book : 'Front Tracking From Conservation Laws' by Holden & Risebro; I have gone through the following calculation: " $\int_{\mathbb R} ...
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1answer
19 views

Proof that the set of integrable real-valued functions is a vector space

From Folland's Real Analaysis: Modern Techniques and Applications: Proposition: Let $(X,\mathcal{M},\mu)$ be a fixed measure space. The set of integrable real-valued functions on $X$ is a real ...
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1answer
79 views

Question about notion $d\mu = fdv$ in Real Analysis of Folland

I'm reading the book Real Analysis of Folland, chapter 3 about signed measure, and there's some notion that confused me. In this book, he defines that $dv = fd\mu$ if $v(E) = \int_E{fd\mu}$, and ...
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0answers
46 views

Suppose $f $ is absolutely continuous and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. [closed]

Problem: Suppose $f: \Bbb R \rightarrow \Bbb R$ is absolutely continuous on every interval $[a,b]$, and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. ...
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1answer
21 views

Integrability of Riesz potential

Given $f\in L^1(\mathbb{R}^3)$, define $$\phi(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|y-x|}\,dy.$$ I was able to show that $\phi$ exists for almost all $x$ (I used the Lebesgue differentiation theorem). ...
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2answers
57 views

Why can I use Fubini' theorem on this function?

I used the fact that $\displaystyle \int_0^\infty\int_0^1 e^{-y}\sin(2xy)\,dxdy=\int_0^1\int_0^\infty e^{-y}\sin(2xy)\,dydx$ to solve $\displaystyle\int_0^\infty e^{-y}\frac{\sin^2(y)}{y}\,dy$. (The ...
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0answers
20 views

Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
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1answer
42 views

The pointwise limit of a measurable function is still measurable?

This is a previous discussion but I just found I didn't get the answer I want... The question is as follows: Assume a sequence of Radon measure $\mu_n\to\mu$ in weak star sense. The domain of $\mu$ ...
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1answer
20 views

Measurability of a version of a random variable

If $X$ is a ($\mathcal{F}$-measurable) random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and $Y$ is a version of $X$ in the sense that $\mathbf{P}(X \ne Y) = 0$ and ...
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1answer
26 views

Compactness of signed measure?

This idea never comes to me but I just realize that I am making a serious mistake that the space of finite signed measure is weakly compact... We all know that the space of finite Radon measure is ...
2
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0answers
45 views

If equality of dual space of a Banach spaces implys the equality of pre-duals?

Assume $ X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them ...
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2answers
18 views

Hausdorff dimension of graph of function

This question came up on an exam Decide the Hausdorff dimension of the graph of the following function for $x>0$ $$y = \log(1+x)\sin\frac{1}{x}$$ In the course, we only touched upon the subject ...
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1answer
67 views

Background for Graduate Real Analysis I Class [on hold]

This semester, I have signed up for a graduate Real Analysis I course (really a course in measure theory/Hilbert Spaces/Lebesgue integration) and have thus far attended two lectures. However, from ...
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1answer
52 views

If $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$

Show that if $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$ Hint: Use Hölder's inequality. But I can't see where I should use it. I'm trying to use it in $\displaystyle\int |f|^p\,d\mu = ...
1
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1answer
22 views

Lusin property (N) for functions of several variables

I just read in a paper by Martio and Zeimer$^1$ that smooth functions ($C^1$) of several real variables have the have the Lusin property (N). I have two questions. First, could someone give me a ...
3
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1answer
37 views

If $f_n\to f$ in measure and $\mu(|f_n|^p)$ is bounded then $\mu(|f|^p)$ is finite

-> The sequence $(\int|f_n|^p\,d\mu)_{n \in \Bbb N}$ is bounded. -> $f_n\to f$ in measure. Prove that f is p-integrable. I'm trying to use the dominated convergence theorem. But I can't find an ...
3
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0answers
54 views

Is the completion of a measure space necessary?

Most important theorems in measure theory do not assume the completeness of measure spaces. Monotone convergence theorem, Dominated convergence theorem, and Fubini's theorem, to name a few. So I ...
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votes
1answer
28 views

Conjecture about regular Borel measures and dense sets with no interior

Suppose that $(X,\tau)$ is a topological space and let $\mathscr B$ denote the Borel $\sigma$-algebra on it. Moreover, let $\mu:\mathscr B\to[0,\infty]$ be a regular Borel measure, that is, ...
1
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1answer
35 views

Help verify a solution showing $f\left(x \right)=\int_\Bbb{R} {{\chi _A}\left(y \right){\chi _B}\left( {x-y} \right)dy} $ is well-defined everywhere

The question is, Let $A,B⊂[0,1]$ be measurable sets with $|A|>1/2$,$|B|>1/2$ where $|*|$ denotes Lebesgue measure. Prove that a. $|A⋂(1-B)|>0$ where $1-B≔{1-x:x∈B}$ and conclude that ...