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

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convergence in distribution in Banach spaces

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
4
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1answer
16 views

Theorem 2.22 from RCA Rudin

I read this interesting result from Rudin's book but I would like to clarify some confusing moments. As I understood $(\mathbb{R}^1, +)$ is group and $(\mathbb{Q}, +)$ is subgroup. He considers ...
2
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1answer
33 views

Integrals over subset of measure space

Let $(X, \mathcal{M}, \mu)$ be a measure space. Suppose $E \in \mathcal{M}$ and $f \in L^+$ where $L^+$ is a space of measurable functions from $X$ to $[0, \infty]$. $\int_E f$ is defined by $\int_X f\...
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1answer
26 views

Real Analysis, Folland Problem 2.4.42, counting measure with convergence in measure

Problem 2.4.42 - Let $\mu$ be counting measure on $\mathbb{N}$. Then $f_n\rightarrow f$ in measure if and only if $f_n\rightarrow f$ uniformly. Attempted proof - Suppose that $\mu$ is a counting ...
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1answer
17 views

Can anyone explain an example related to vague convergence?

Let {$X_n$} be a sequence of random variable and {$\mu_n$} be a sequence of measures induced by {$X_n$} such that $\mu_n(B) = \mathbf{P}(X_n^{-1}(B))$. Suppose $X_n$ following a uniform distribution ...
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2answers
42 views

Real Analysis, Folland Problem 2.4.33 Modes of Convergence

Background Information: Theorem 2.30 - Suppose that $\{f_n\}$ is Cauchy in measure. Then there is a measurable function $f$ such that $f_n\rightarrow f$ in measure, and there is a subsequence $\{...
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2answers
34 views

Real Analysis, Folland Proposition 2.30 Modes of Convergence

Proposition 2.30 - Suppose that $\{f_n\}$ is Cauchy in measure. Then there is a measurable function $f$ such that $f_n\rightarrow f$ in measure, and there is a subsequence $\{f_{n_j}\}$ that converges ...
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0answers
26 views

Prove $X_n \nrightarrow X = \bigcup_{k=1}^{\infty} \{|X_n - X| > \frac{1}{k}\}$ [on hold]

Probability with Martingales: Important inequalities: 1, 2 $$\liminf x_n > z \implies x_n > z \ \text{eventually}$$ $$\liminf x_n < z \implies x_n < z \ \text{infinitely ...
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1answer
83 views

liminf of sequence of iid random variables

if $P(\liminf (X_n>a))=0$, does that mean $\liminf X_n <a$ almost surely? Where $X_n$ are iid random variables.
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1answer
27 views

A graph of any function has content zero?

I know that if $f$ is integrable them his graph have content zero, but if we consider $f : \mathbb{R} \rightarrow \mathbb{R}$ as $f(x) = 1$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x \in \mathbb{R} - \...
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Total Variation distance between two Poisson distributed measures.

Let $\mu$ have the Poi(1) and $\nu$ have the Poi(2) distrbutions. What is the total variation of these two? So the total variation is definetd as : $|\mu-\nu|_{TV}=\max_{A\subset S}|\mu(A)-\nu(A)|=1/...
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1answer
13 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
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0answers
50 views

how to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let D be a domain of $\Bbb R^n$ anf $f_n$ : D $\to$ $\Bbb R$. Say if $f_n \in L^1(D)$ First of all I need to check that both $f_n$ anf $...
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2answers
27 views

Does the collection $\{ \ [n, \infty) \ \mid \ n \in \Bbb N\}$ generate the Borel $\sigma$-algebra on $\Bbb R$?

Does the collection $\mathfrak C = \{ \ [n, \infty) \ \mid \ n \in \Bbb N\}$ generate the Borel $\sigma$-algebra on $\Bbb R$? I came across this problem in a paper. It seems like the answer is No. ...
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1answer
35 views

Uniform integrability of a sequence of random variables defined by a recursive relation

I have an i.i.d sequence $(u_j)_{j\in \mathbb{Z}_+}$ with zero mean and finite variance, say $\sigma^2$. Furthermore, I have another random variable $X_0$ (defined on the same probability space) which ...
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2answers
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Real Analysis, Folland 2.33 Egoroff's Theorem Modes of Convergence

2.33 Egoroff's Theorem - Suppose that $\mu(X) < \infty$, and $f_1,f_2,\ldots$ and $f$ are measurable complex-valued functions on $X$ such that $f_n\rightarrow f$ a.e. Then for every $\epsilon > ...
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If $Q$ is a cube in $\Omega$, then $m(\varphi(Q))\geq\int_{Q}{|J(x)|dx}$

I wanna prove the theorem of CHange of variables: Let $\varphi:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{n}$ a regular diffeomorphism $C^{1}$ over the open $\Omega$, and let $f:\varphi(\Omega)\to\...
3
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1answer
17 views

Measure on Borel $\sigma$-algebra of $\mathbb{R}$ is Lebesgue-Stieltjes measure

Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0, x])$ if $x \ge 0$ and $\alpha(x) = -\mu((x, 0])$...
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2answers
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Is $\bigcup_{x \in A} [x - 1, x + 1]$ Lebesgue measurable, where $A$ is a Lebesgue measurable subset of $\mathbb{R}$?

Suppose $A$ is a Lebesgue measurable subset of $\mathbb{R}$ and $$B = \bigcup_{x \in A} [x - 1, x + 1].$$Is $B$ Lebesgue measurable?
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1answer
28 views

Does there exist $G$ open and $F$ closed such that $F \subset A \subset G$ and $m(G - F) < \epsilon$?

Let $m$ be Lebesgue measure and $A$ a Lebesgue measurable subset of $\mathbb{R}$ with $m(A) < \infty$. Let $\epsilon > 0$. Does there exist $G$ open and $F$ closed such that $F \subset A \subset ...
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2answers
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Preimage of open set is Lebesgue measurable only if the function itself is measurable

It is a simple result in my book saying the proof is trivial, but I can not seem to show it. If someone can provide a hint just to help me begin my proof, it would be of assistance. Assume you know ...
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1answer
16 views

Lebesgue outer measure equals Lebesgue Inner Measure

Definition. (Lebesgue Measurable) A set $E$ is said to be Lebesgue measurable if there exists an open set $G$ and a closed set $F$ such that $F\subset E\subset G: m^*(G\setminus F)<\epsilon$. ...
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0answers
25 views

Convergence of measures in total variation sense

Suppose we have measures that defined over the set $\{0,1,2,..,C\}$. Let $\{\mathbb{P}_{n,m}\}$ be a sequence of measures. Suppose that for fixed $n$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}_n$ ...
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0answers
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$\mu * \nu$ a finite Borel measure in $\mathbb{R}$?

Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb{R}$. For any Borel set $A \subset \mathbb{R}$, define$$\mu * \nu(A) = \mu \times \nu(\{(x, y) \in \mathbb{R}^2 : x + y \in A\}).$$Is $\mu * ...
2
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2answers
27 views

$\{A \subset X: \chi_A \in \mathcal{F}\}$ is a sigma algebra

Suppose $\mathcal{F}$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal{F}$ and $f + g$, $fg$, and $cf$ are in $\mathcal{F}$ whenever $f$, $g \in \...
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1answer
14 views

Ascending chain of monotone classes, $A$ necessarily in $\mathcal{M}$

Suppose $\mathcal{M}_1 \subset \mathcal{M}_2 \subset \ldots$ are monotone classes. Let $\mathcal{M} = \bigcup_{n = 1}^\infty \mathcal{M}_n$. Suppose $A_j \uparrow A$ and each $A_j \in \mathcal{M}$. Is ...
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35 views

Is this $\sigma$-algebra necessarily uncountable or not? [on hold]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
3
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0answers
31 views

Is $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$ a $\sigma$-algebra of subsets of $X$ or not?

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define$$\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}.$$Is $\mathcal{B}$ a $\...
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2answers
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Real Analysis, Folland Proposition 2.29 Modes of Convergence

Background Information: $f_n\rightarrow f$ in $L^1$ $\Leftrightarrow$ $\forall\epsilon > 0,\exists N$ $\forall n\geq N$ $\int |f_n - f| < \epsilon$ A sequence $\{f_n\}$ of measurable complex-...
2
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0answers
25 views

If $\mu(E)\geqslant 0$ is it true that $E\in \mathfrak{M}$?

Suppose $(X,\mathfrak{M},\mu)$ be a mesure space. Let $E$ such that $\mu(E)\geqslant 0$. Can we conclude that $E\in \mathfrak{M}$? I think YES because $\mu$ is the set function with domain $\mathfrak{...
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2answers
18 views

Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
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0answers
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Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
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0answers
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Construction of Lebesgue measure in Rudin's RCA book

This theorem from Rudin's RCA book. Here's one moment from it's proof which seems to me very weird. Rudin states that equality $\lambda(E)=m(E)$ holds for all Borel sets. But I think that it's ...
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0answers
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Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
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0answers
35 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
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2answers
52 views

Independent $\sigma$-algebras using $\pi$-$\lambda$-theorem

Let $\mathcal{E}_1, ...,\mathcal{E}_n$ be collections of measurable sets on $(\Omega,\mathcal{F},P)$, each closed under intersection. Suppose \begin{align*} P(A_1\cap...\cap\ A_n)=P(A_1)\cdot ... \...
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1answer
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Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
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1answer
23 views

Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
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0answers
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If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
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1answer
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+100

Diffuse-like decomposition of the segment $[0,1]$ in accordance with Lebesgue measure

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any subsegment $[a,b]\...
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1answer
18 views

Mensuration- Cubes [on hold]

A cube of 64 cubic ft is cut with a plane passing through two diagonally opposite edges.What is the increase in total surface area of the two pieces over that of the original cube?
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The finite-dimensional distribution of a stochastic process

Let $K(s,t)$ be a real function over $T\times T$, where $T$ is arbitrary. $K$ has two properties: $K$ is symmetric ($K(s,t)=K(t,s)$). $K$ is nonnegative-definite ($\sum_{i,j=1}^k K(t_i,t_j)x_ix_j\...
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1answer
38 views

Real Analysis, Folland Theorem 2.26 Integration of Complex Functions

Background information: Theorem 2.10 - Let $(X,M)$ be a measurable space. a.) If $f:X\rightarrow [0,\infty]$ is measurable, there is a sequence $\{\phi_n\}$ of simple functions such that $0 \...
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0answers
50 views

Kolmogoroff 0-1 does this proof work?

I have thought at this proof of the Kolmogorov 0-1 Law varying a little the sketch found in Probability essentials (Jean Jacod, Philip Protter). My questions are Is it a valid proof? Is it a bad ...
4
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1answer
61 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
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1answer
15 views

Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
2
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1answer
38 views

Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
4
votes
1answer
124 views

Weak Law of Large Numbers

The Weak Law of Large Numbers is often stated with the iid assumption for the underlying RV's. However, I have seen the independence assumption being diluted to the "uncorrelatedness" assumption (e.g.,...
3
votes
2answers
41 views

Application Banach-Alaoglu Theorem

When reading about Banach-Alaoglu Theorem on Wikipedia, I read the following assertion: '' Let $f_n$ be a bounded sequence of functions in $L^p$. Then there exists a subsequence $f_{n_k}$ and an $f\...
3
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1answer
47 views

If $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|\max (0, \log |f_n|)<\infty$, then $f_n\to f$ in $L_1$

I'm going through old analysis qualifying exams, and have come to a roadblock on the following problem: Suppose that $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|...