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

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Exercise $112Y(d)$ in Measure Theory by Fremlin

Let $(X, \Sigma)$ be a measurable space and $N$ be a collection of measures each with domain $\Sigma$. Let: $$\mu E = \inf \left\{ \sum_{i=1}^m \nu_i(E \cap F_i) \mid \text{ $(F_i) \subseteq \...
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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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+50

$\mu * \nu$ 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 $\mu * ...
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2answers
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$\{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
12 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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Is this $\sigma$-algebra necessarily uncountable or not?

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 ...
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1answer
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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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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-...
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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
16 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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18 views

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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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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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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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 ...
3
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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
25 views

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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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 ...
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Measurable isomorphism between two non-totally ergodic systems

Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
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1answer
22 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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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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Prove $X_n \nrightarrow X = \bigcup_{k=1}^{\infty} \{|X_n - X| > \frac{1}{k}\}$ [on hold]

Probability with Martingales: Important inequalities: 1, 2 $$\liminf x_n > z \to \liminf(x_n > z)$$ $$\liminf x_n < z \to \limsup(x_n < z)$$ What I tried: I think the ...
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1answer
82 views
+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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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 ...
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1answer
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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 \...
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1answer
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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:...
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1answer
123 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.,...
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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
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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|...
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1answer
39 views

Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
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1answer
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Continuity from below and above

In Folland's Real analysis, two of properties of measures are stated as follows: Let $(X,\mathcal{M}, \mu)$ be a measure space. Continuity from below: If $\{E_j\}_1^{\infty} \subset \mathcal{M}$ ...
4
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1answer
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Conditions on a complex measure to be real

Let $(X,\mathcal{S}, \mu)$ be a measure space with $X$ a locally compact Hausdorff space, $\mathcal{S}$ the Borel subsets of $X$ and $\mu$ a complex measure. Suppose that $$ \int_X f \ d\mu \in \...
2
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1answer
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Real Analysis, Folland Theorem 2.25 Integration of Complex Functions

Theorem 2.25 - Suppose that $\{f_j\}$ is a sequence in $L^1$ such that $\sum_{1}^{\infty}\int |f_j| < \infty$. Then $\sum_{1}^{\infty}f_j$ converges a.e. to a function in $L^1$, and $$\int \sum_{1}^...
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$\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ - definition?

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
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Kolmogorov 0-1 Law Converse?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. ...
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Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
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1answer
32 views

Vainberg Theorem in measure theory

In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let $(f_n)$ a sequence in $L^{p}(\Omega)$ and $f \in L^{p}(\Omega)$, such that $f_{n} \rightarrow f$ in $L^{p}(\...
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Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
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1answer
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Ash and Doleans-Dade Probability and Measure Theory Section 1.2 Question 2

Ok so in section 1.2 of chapter 1, the authors pose the following challenge: Let $\mu$ be the counting measure on $\Omega$, where $\Omega$ is an infinite set. Show that there is a sequence of sets $...
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1answer
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Real Analysis, Folland Problem 2.3.19 Integration of Complex Functions

Problem 2.3.19 - Suppose $\{f_n\}\subset L^1(\mu)$ and $f_n\rightarrow f$ uniformly. a.) If $\mu(X) < \infty$, then $f\in L^1(\mu)$ and $\int f_n \rightarrow \int f$. b.) If $\mu(X) = \...
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Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)?

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)? This is from the last sentence in the proof in the following ...
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19 views

Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
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1answer
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Real Analysis, Folland Proposition 2.21 Integration of Complex Functions

Proposition 2.21 - The set of integrable real-valued functions on $X$ is a real vector space, and the integral is a linear functional on it. Attempted proof - Note that we can derive the axioms of a ...
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1answer
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Real Analysis, Folland The Dominated Convergence Theorem

Background Information: Proposition 2.16 - If $f\in L^+$, then $\int f = 0$ iff $f = 0$ a.e. Question: 2.24 The Dominated Convergence Theorem - Let $\{f_n\}$ be a sequence in $L^1$ such that ...
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0answers
21 views

Alternative Proof about Product Measures and Iterated Integrals

Background Theorem 2.36 of Folland's Real Analysis says that if $(X,M,\mu)$ and $(Y,N,\nu)$ are sigma finite measure spaces, and $E\in M\bigotimes N$, then $x\mapsto \nu(E_x)$ and $y\mapsto \mu(E^y)$ ...
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Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...