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

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Green identity for measures with compact support

Let $\Omega\subset\mathbb{R}^N$ be a bounded, smooth domain. Assume that $\mu \in \mathcal{M}(\Omega)$ has compact support in $\Omega.$ Let $u\in W_0^{1,1}(\Omega)$ be a solution of $$ \left\{ ...
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1answer
14 views

quick question on measurability of random variables and what becoming a deterministic function means.

we stated a theorem in class: if X r.v. is $\sigma(Y)$ measurable then X is a function of Y, where $\sigma(Y)$ signifies the sigma algebra of Y. This is fine. The Professor sometimes states that X ...
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1answer
18 views

Show that $A_f$ is measurable set

Let $(X,\mathfrak M,\mu)$ be space with \sigma finite measure and let $f$ be positive measurable function on $X.$ If $\lambda=\mu\times m$, where $m$ is Lebesgue measure, show that $A_f=\{(x,t):x\in ...
2
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1answer
38 views

Does Vitali set imply the axiom of choice

I know that the construction of Vitali set needs the axiom of choice, but this only states that $AC \implies V$. Is it also true that $V \implies AC$? If $\neg AC \implies \neg V$, then what ...
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0answers
13 views

Are two distinct Lebesgue Stieltjes measures have the same domain?

Let $F,G:\mathbb{R}\rightarrow \mathbb{R}$ be a monotonically increasing right continuous function. Define $\mu_F((a,b])=F(b)-F(a)$ for all $a,b\in\mathbb{R}$ such that $a≦b$. Let $\mu^*_F$ be the ...
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1answer
98 views

Help with Borel sets

I have to prove that if $E$ is a set such that $\mu (E)=1$, then, there is a subset $E_t\subset E$ such that $\mu(E_t)=t$ for every $0\leq t\leq 1$ I think, that given a measurable set $E$ there is a ...
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0answers
84 views

Spivak Problem 3-23

The problem reads, with $A \subset \mathbb{R}^n$ and $B \subset \mathbb{R}^m$ for some $n, m \in \mathbb{Z}^+$: Let $C \subset A \times B$ be a set of content $0$. Let $A' \subset A$ be the set of ...
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2answers
77 views

Formally show that the set of continuous functions is not measurable

Let $C(\mathbb{R})=\{ f:\mathbb{R}\to \mathbb{R} \colon \ f \text{ continuous}\}\subseteq \mathbb{R}^{\mathbb{R}} $. How to prove formally that $C(\mathbb{R}) \notin ...
3
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1answer
48 views

Integration using Lebesgue dominated convergence theorem

This is an old comp question I'm working on. $$\lim_{n\to\infty}\int_{[0,1]}\frac{d\lambda}{x^\frac{1}{n}(1+\frac{x}{n})^n}$$ I am having trouble finding a dominating function. Thinking about the ...
2
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1answer
34 views

Product measure and integrals of simple functions

Let $(\Omega_1 , \mathcal{X}, \mu)$ and $(\Omega_2 , \mathcal{Y}, \nu)$ be two $\sigma$-finite measure spaces, and let $\mu \times \nu$ be product measure on the $\sigma$-algebra $\mathcal{X} \times ...
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0answers
76 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
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0answers
17 views

Let S be the set of of all step functions on [0, 1] with rational range and rational partition points.

Hello need help with this problem: Let $S$ be the set of of all step functions on $[0, 1]$ with rational range and rational partition points. 1- Show that the closure of $S$ in $L^\infty[0,1]$ ...
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1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
2
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1answer
54 views

can a compact set have infinite measure?

Can a compact set have infinite measure? It does not seem to violate the measure axioms. This is not true in the case of Lebesgue measure. So I am also wondering is there any clean cut condition for ...
2
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1answer
28 views

Radon–Nikodym theorem: special case

Let $X$ be a locally compact Hausdorff space with the Borel $\sigma$-algebra $\mathscr B_X$. Suppose that $\mu$ is a positive measure, $\nu$ is a finite positive measure, and $\nu\ll\mu$. It is known ...
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3answers
48 views

Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
4
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1answer
58 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
2
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1answer
22 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
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1answer
15 views

Infimum of an outer measure

I'm stuck in this problem. Let I a set of index and $m_i$ outer measure for all $i\in I$. Show that $\inf\limits_{i\in I}m_i$ is an outer measure. Specifically I don't know how to show the ...
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1answer
40 views

Find $\liminf X_n$ where $X_n=1_{[n,n+1]}$?

My attempt: Suppose $\omega=n_0$. Then choose $N\geq n_0+1$.Threfore, $X_N(\omega)=0$. Therefore, $\inf_{k\geq N}X_k(\omega)=0$. Does it suffice to prove that $\liminf\limits_{n \rightarrow \infty} ...
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2answers
61 views

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials?

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials? And why? I know it is the uniform limit on a set take out some finite measurable set but not sure if I can say more. Thanks.
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1answer
17 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
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0answers
25 views

A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu $ and $\lambda $ are measures in $M$ st $\mu \ge $ $\lambda $ then show that $m$ defined as $\mu= \lambda + m $ is a measure 2) Prove that if ...
1
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1answer
19 views

Prokhorov-like convergence

Let $(X,d)$ be a metric space, and for any $A\subseteq X$ define $$ A^\delta:=\{y\in X:\exists x\in A \text{ such that }d(x,y)\leq \delta\}. $$ Under which conditions on $(X,d)$, $A \subseteq X$ and ...
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1answer
28 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
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1answer
27 views

How to show $\displaystyle s=\sum_{i=1}^n \alpha_i \chi_{A_i}$ measurable implies $A_i$ measurable for all $i=1, \ldots, n$?

Let $X$ be a measurable set and $s:X\longrightarrow [0, \infty)$ be a simple function. It is easy to see $$s=\sum_{i=1}^n \alpha_i \chi_{A_i},$$ where $\{\alpha_1, \ldots, \alpha_n\}$ is the set of ...
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7answers
292 views
+50

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
4
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1answer
85 views

Measures $\mu$ such that $\mu(a+A)\leq c\ \mu(A)$

Let $\mu$ be a positive measure on $\mathbb{R}$ such that $\mu[a,b]<+\infty$, for all $a,b\in\mathbb{R}$ and $\mu(\mathbb{R})=+\infty$. The set $a+A$ denotes the translation set of $A$ by a, i.e. ...
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2answers
48 views

Inclusion for pairwise disjoint sets and $\limsup A_n = \emptyset$

Spin-off from here. 1 Please give an example of how the following does not hold for a collection that is not pairwise disjoint. $$ \bigcup_{k \ge n+1} A_k = A\setminus (A_1 \cup\cdots \cup A_n) $$ ...
1
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0answers
53 views

$X_n=n^21_{(0,1/n)}$. What is $\limsup\limits_{n \rightarrow \infty} X_n$

$X_n=n^21_{(0,1/n)}$. What is $\limsup\limits_{n \rightarrow \infty} X_n$? What about its $\liminf\limits_{n \rightarrow \infty} X_n$? My attempt: For each $n$ on $\{0, [1/n,1]\}$, we have ...
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0answers
38 views

Prove that E$(\liminf\limits_{n\rightarrow \infty} X_n )\leq \liminf\limits_{n\rightarrow \infty}E(X_n)$

I want to prove that if $X_n\geq 0$, then E$(\liminf\limits_{n\rightarrow\infty} X_n )\leq \liminf\limits_{n\rightarrow\infty}E(X_n)$. My attempt: $E(\liminf\limits_{n \rightarrow \infty} ...
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0answers
7 views

Extension of premeasure to outer measure

I'm stucked in this problem. Show that all outer measure $\mu_*$ can be expresed of the form $$\mu_*(A)=\inf_{A\subset\cup C_i}\sum \tau(C_i)$$ where $\tau$ is a premeasure of a colection of sets ...
2
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1answer
29 views

Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?

Let $C$ is a compact subset of $\mathbb R,$ $V\subset \mathbb R,$ and $0<m(V)<\infty,$ where $m$ is a Lebsgue measure on $\mathbb R.$ My Question is: Can we expect to find $k\in ...
4
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0answers
48 views

Jordan decomposition of linear functionals

Let $X$ be a locally compact Hausdorff space. Also, let $C_0(X,\mathbb R)$ denote the vector space of such continuous functions $f:X\to\mathbb R$ that the set $\{x\in X\,|\,|f(x)|\geq\varepsilon\}$ is ...
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1answer
25 views

convergence in measure of min $(f_n,g)$

I was reading a proof of a convergence in measure variant of fatou's lemma earlier and there was a seemingly easy part of it I just could not verify. Assume $(f_n)_{n \in \mathbb N}$ is a sequence of ...
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1answer
25 views

Almost Trivial $\sigma-$fields

I am trying to understand the proof of the following Lemma form the book A probability path by sidney Resnick. Lemma: Let $\mathcal{G}$ be an almost trivial $\sigma-\text{field}$ and let $X$ be a ...
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4answers
99 views

Are null sets necessarily closed?

Hi everyone: Is a null set of $\mathbb{R}^n$, $(n>0)$, necessarily closed? Give a counter example. Thanks for your reply.
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1answer
30 views

Complex measures vs. Positive Measures

In his real and complex analysis, Rudin writes that the right hand side of the expression $\mu(E) = \Sigma \mu(E_i)$ must necessarily converge for any countable partition $\{E_i\}$ of a measurable E, ...
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2answers
42 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
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2answers
37 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
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1answer
121 views

Measure which does not grow faster than Lebesgue

Is there an example of a measure $\mu$ on $\mathbb{R}$ which is not absolutely continuous with respect to Lebesgue measure such that $\mu[\mathbb{R}]=+\infty$ but $$\limsup_{a\to ...
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1answer
24 views

Existence of a sequence of continuous functions converging pointwise to a characteristic function.

I'm reading Rudin's Real and Complex Analysis and in section 5.11 he makes the next assertion: Put $g_n(t)=1$ if $D_n(t)\geq 0$, $g(t)=-1$ if $D_n(t)<0$. There exist $f_j\in C(T)$ such that ...
7
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1answer
32 views

If $w$ is in weak $A_{\infty}(d\mu)$ where $d\mu$ is a doubling measure, then is $w\,d\mu$ doubling?

Let $\mu$ be a positive Borel measure on $\mathbb{R}^n$ and let it be doubling i.e. there exists a a constant $C>1$ such that $\mu(B(x_0, 2r)) \leq C \mu(B(x_0,r))$ for all balls $B(x_0,r)$. Let ...
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1answer
71 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
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0answers
33 views

Limit of measurable function is measurable

This question has been asked already here but I didn't get a satisfactory solution and didn't want to bring up an old question. Here is the question : Let $\{f_n\}$ be a sequence of measurable ...
2
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1answer
33 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
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2answers
40 views

The applicability of the Dominated Convergence theorem on the real line

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it ...
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1answer
22 views

Convergence in Probability implies Weak Convergence Proof Question

I'm trying to follow a proof for showing $\displaystyle \lim_{n\rightarrow \infty} P[|X_n-X|>\epsilon] = 0 \Rightarrow X_n \rightarrow_p X$ The first step of the proof says: $P[X \leq x-\epsilon] ...
1
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1answer
36 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
0
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0answers
25 views

Measure Theory - working with unusual measures and set functions

Let $m$ define the Lebesgue measure. Let $\mu$ define the measure $\mu(A)=m(A\cap(0,1))$ for a Borel set $A$. Let $K=\bigcap \{A:A$ is closed, $\mu(A)=1\}$, $D=\bigcap \{G:G$ is open, ...