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

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26 views

Improbably doesn´t mean impossible

I have a question: in the past days I give myself the question: given a measurable space $(\Omega,\mathcal{F},\mu)$, what kind of conditions about $\mathcal{F}$ and $\mu$ we need for: The only ...
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1answer
21 views

a proposition in the construction of the Borel measures on the real line

In the construction of the Borel measures on the real line, the following proposition is used in Folland's Real Analysis: Here is my question: If one replaces $(a_j,b_j]$ with $[a_j,b_j)$, ...
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1answer
53 views

Two questions on measurable sets.

I'm learning about measure theory, specifically measurable sets, and need help with the following two questions: $(1)$ For $n \in \mathbb{N}$, let $E_n = \{x \in [0, 2\pi] : \sin x < {1 \over n}...
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1answer
30 views

Show that $\mu(E \cup A) + \mu(E \cap A) = \mu(E) + \mu(A)$

Let $E$ be a measurable subset of $X$, then show that for every subset $A$ of $X$ the following equality holds: $$\mu(E \cup A) + \mu(E \cap A) = \mu(E) + \mu(A).$$ I know since $E$ is measurable $$\...
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0answers
38 views

Integration w. r. t. counting measure

I'm learning about measure theory, specifically integration w.r.t. counting measure, and need help to verify my understanding of this new notion through two exercises. (1) Let $(\mathbb{N},\scr{P}...
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1answer
24 views

How do I show that $\mu$ is a measure?

Let $S$ be a semiring, and let $\mu: S \to [0,\infty]$ be a set function such that $\mu(A) < \infty$ for some $A \in S$. If $\mu$ is $\sigma$-additive, then show that $\mu$ is a measure. What ...
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0answers
23 views

Hilbert space mean ergodic theorem application

Let $(u_n)_{n \geq 0}$ be a bounded sequence in a Hilbert space. We define $$ s_h = \limsup \frac 1 N \sum_{n=o}^{N-1} \langle u_{n+h} , u_n \rangle $$ Show that, if $ \lim \frac 1H \sum_{h=o}^{H-1} ...
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27 views

How to prove this identity? Transformation theorem

Let $A\in\mathbb R^n$ be a measurable set with finite measure. For a fixed vector $p\in\mathbb R^{n+1}$ define a cone with basis $A$ and peak $p$ as $$K(A,p)=\{tp+(1-t)q \in\mathbb R^{n+1} \,| \, q \...
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2answers
55 views

Measure-preserving mapping

Let $(X, \mu, T)$ be a mesure-preserving mapping. Let $A \subset X$ be a measurable subset such that any point in $A$ eventually comes back to $A$. We define space $(A, \mu_A)$, $ \mu_A ( B) = \mu (B) ...
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31 views

Strict inequality in Fatou's lemma

Put $f_n={1}_E$ if $n$ is odd, $f_n=1-1_E$ if $n$ is even. What is the relevance of this example to Fatou's lemma? Proof: We see that $\int \limits_{X}f_nd\mu= \mu(E^c)$ if $n$ is even and $\int \...
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1answer
35 views

Doob Decomposition Theorem - submartingale iff increasing

Probability with Martingales To prove $b$ I tried: $$A_n \ge A_{n-1}$$ $$\iff E[X_{n} - X_{n-1} | \mathscr F_{n-1}] \ge 0$$ $$\iff E[X_{n} | \mathscr F_{n-1}] \ge X_{n-1}$$ That ...
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1answer
23 views

Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability?

Probability with Martingales This is my understanding of what is going on in the proof above: We first assume $X$ has such Doob Decomposition in order to figure out what $A$ to use in ...
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1answer
104 views

Almost sure convergence of equal weighted sum

Let $Z_1, Z_2, ...$ be independent random variables in the same probability space defined as follows: $$P(Z_n=n)=P(Z_n=-n)=\frac{1}{2n^2} \space \mathrm{and} \space P(Z_n=0)=1-\frac{1}{n^2}$$ Is it ...
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1answer
31 views

Measure on torus invariant under multiplication

Let $T: [0,1] \rightarrow [0,1]$ be a multiplication by $ \beta >1$ mod $1$. Show that $h(x) d x$ is $T$-invariant where $$h (x) = \sum_{n \geq 0} \beta^{-n} \chi _{[0,T^n (1)]} (x)$$ ($\chi$ is ...
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1answer
24 views

Determine the two $\sigma$-algebras of subsets of $X$ generated by

Let $A$ be a fixed subset of a set $X$. Determine the two $\sigma$-algebras of subsets of $X$ generated by $\{A\}$ $\{B: A\subseteq B \subseteq X\}$ I managed to find the first one, ...
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2answers
46 views

The set of point where limit function is finite

Prove that the set of points at which a sequence of measurable real-valued functions converges (to a finite limit) is measurable. Proof: Let $f_n(x):X\to \mathbb{R}$ be a sequence of real-valued ...
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1answer
73 views

Weak convergence with uniformly convergent functions.

Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that $$ \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\...
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0answers
44 views

Prove the series $\sum_{n=1}^\infty \frac{1}{n^2} g(x-q_n)$ is absolutely convergent almost everywhere; $g(x)=x^{-1/3}$.

Let $\mathbb{Q} = \{q_n:n\in \mathbb{N}\}$ and $g(x) = \begin{cases} x^{-\frac{1}{3}}& x \neq 0\\ 0& x=0 \end{cases}$. I can't figure out how to prove the series $\sum_{n=1}^\infty \frac{1}{n^...
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2answers
48 views

Real Analysis, Folland Theorem 1.18 Borel measures on the real line

Background information - We fix a complete Lebesgue-Stiltjes measure $\mu$ on $\mathbb{R}$ associated to the increasing right continuous function $F$, and we denote by $M_{\mu}$, the domain of $\mu$. ...
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2answers
81 views

Introduction to Measure Theory, length of an Interval.

Let $I$ be an interval on real line, (in the form of $(a, b)$,$[a,b)$,$(a,b]$ or $[a,b]$, where $a,b \in \mathbb R$) and $|I|:= b-a$. Prove that $$ |I| = \lim_{N\to\infty}\frac{\#\left(I\cap\tfrac{\...
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2answers
49 views

Does a set with strictly positive Lebesgue measure contain an interval?

I am studying a function whose Fourier transform is zero on a set of strictly positive Lebesgue measure and I need to know this: If a set has a strictly positive Lebesgue measure can we prove that it ...
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2answers
34 views

Can anyone explain the connection between absolutely continuous functions on R and absolutely continuous measure?

Can anyone explain the connection between absolutely continuous functions on R and absolutely continuous measure? I think the connection is also relevant to radon nikodym derivatives and fundamental ...
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1answer
32 views

If $f_n$ converges to $f$ in $L^1(\mathbb{R})$ and $f_n$ converges to $g$, what relation exists between $f$ and $g$?

Take $(f_n)$ to be a sequence in $L^1(\mathbb{R})$ and suppose it is true that $(f_n)$ converges in $L^1(R)$ to a function $f \in L^1(\mathbb{R})$. Let $g$ be a function such that $(f_n)$ converges to ...
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0answers
12 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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1answer
33 views

Prove that the $\sigma$-sets of the semiring $S$ form a topology for $\mathbb R$.

Prove that the $\sigma$-sets of the semiring $$S=\{[a,b): a,b, \in \mathbb R, a \le b \}$$ form a topology for the real numbers. I know that a subset $A$ of $\mathbb R$ is a $\sigma$-set w.r.t. $S$ ...
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22 views

Probability that there exists $M>0$ such that two processes $\{X_t\}$ and $\{Y_t\}$ are smaller than $M$ at the same time, for infinitely many $t$.

Suppose I have two iid sequences of random variables $\{X_t\}_{t\in\mathbb{N}}$ and $\{Y_t\}_{t\in\mathbb{N}}$, both absolutely continuous with full support. I know that with probability one, for any $...
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34 views

Is distribution of $Y = \sum_{n=1}^{\infty}0.5^{n} X_n$ Lebesgue measure? [closed]

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent binary random variables defined over probability space $(\Omega,\mathcal{A},P)$ such that $P(X_n = 0) = P(X_n = 1) = 0.5$. Define $Y = \sum_{n=...
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2answers
57 views

Suppose $A,B\subseteq [0,1]$ are Lebesgue measurable with measure of at least $1/2$.

Suppose $A,B\subseteq [0,1]$ are Lebesgue measurable with measure of at least $1/2$. Prove there is some $x\in [-1,1]$ such that $\mu((A+x)\cap B)\geq 1/10$. This is a previous qual question. My ...
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1answer
32 views

Can anyone give an example of decomposing a function defined on R into an absolutely continuous part and singularly continuous part?

Can anyone give an example of decomposing a function defined on the real line into an absolutely continuous part and a singularly continuous part?
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1answer
22 views

How to calculate this Lebesgue integral?

Let $(X,A,\mu)=(Y,B,v)=(N,M,c)$ where N is the set of natural numbers, c is the counting measure and M is the power set of N. Define $$f(x,y)= \left\{ \begin{array}{ll} ...
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41 views

Help with change of measure and martingales

Consider two three stochastic processes $X$, $Y$ and $Z$ in probability space $(\Omega, (\mathcal F_t)_{t \geq0},\mathbb P)$ such that $$ X_t = \exp\left(\int_0^t f_s ds\right), $$ $$ Y_t = \exp\...
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1answer
20 views

Show that $S= \{C_1\setminus C_2: C_1, C_2 \text{ closed} \}$ is a semiring.

Let $X$ be a topological space, then show that the collection $$S= \{C \cap O:C \text{ closed and }O \text{ open} \} = \{C_1\setminus C_2: C_1, C_2 \text{ closed} \}$$ is a semiring of subsets of $X$. ...
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2answers
27 views

Random variables and independence of $\sigma$-algebras

If a random variable $X$ is independent from the $\sigma$-algebra $F_t$ for every $t$ in a collection of indexes, is it true that $X$ in independent from the $\sigma$-algebra generated by all the $F_t$...
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1answer
26 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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19 views

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
131 views

What is the motivation behind the arbitrary union topological axiom?

1. Why is the arbitrary union axiom in the definition of topology necessary? 2. Why is it useful? Why might we expect ("intuitively") that it should be useful? 3. What is the (historical) ...
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32 views

How does $\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ not follow by 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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41 views

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
28 views

Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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0answers
21 views

Almost surely interpretation in conditional probabilities

Consider the random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, P)$. Suppose $Y$ is a discrete random variable with support $\mathcal{Y}\subset \mathbb{R}$. Suppose $X$...
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1answer
64 views

Existence of a Lebesgue measurable set

The following is from Carother's Real Analysis: Suppose that $E$ is Lebesgue measurable with $m(E)=1$. Show that there is a measurable set $F\subset E: m(F)=1/2$. Carothers offers a hint which ...
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10 views

scotts theorem and other representation theorems for order aggreeing qualitative quantitative probability measures

Scotts theorem and other theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, finite cancellation axioms ...
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61 views

Difference between $d\mu(x)$ and $\mu(dx)$

In my lecture notes of probability course I found two different notations involving $d,\mu$ and $x$: is there any difference between $\mu(dx)$ and $d\mu(x)$? For example I read $\mu(dx) = \frac{1}{\...
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65 views

Is there always a Minimal Product Measure

I am studying measure theory and I have a question concerning the wikipedia-article "Product measure". I already asked on the Wikipedia-"talk"-page but so far noone answered. The problem concerns the "...
3
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1answer
39 views

Proof that the counting measure of rationals is a measure

I was trying to solve this problem and want to check if my reasoning is correct. Let $\mu$ be a measure on $\mathcal{B}(\mathbb{R})$ such that $\mu(A):=\#\{q, q \in \mathbb{Q}\cap A\}$, prove that $\...
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1answer
52 views

Can Lebesgue Dominated Convergence always be used?

Suppose I want to find the derivative $$\frac{d}{dx}\int f(x,y) dy.$$ I want to know under what condition it would be equal to $$\int \frac{d}{dx}f(x,y) dy.$$ Of course, if I can find a suitable ...
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1answer
56 views

Interchanging Expectation and Derivative

Suppose I have a random function, $f(x)(\omega)$. And that for fixed $\omega$, we have the derivative $g(x)(\omega)=\frac{d}{dx}f(x)(\omega)$. For a fixed $x$, I can find the expectation $E(f(x))$. ...
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1answer
38 views

Real Analysis Folland Proposition 1.13

Proposition 1.13 - If $\mu_0$ is a premeasure on $\mathcal{A}$ and $\mu^*$ is defined by (1.12) then a.) $\mu^*|\mathcal{A} = \mu_0$ b.) every set in $\mathcal{A}$ is $\mu^*$-measurable ...
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1answer
73 views

Any borel measure on a locally compact, sigma-compact, Hausdorff space is regular?

on the book "measure and integration theory" by Heinz Bauer , theorem 29.12, the same proof works for a sigma-compact space instead of a locally compact space with countable basis(which is in ...