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

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Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
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1answer
214 views

Connection between separable measure spaces and $\sigma$-finite measure spaces

I recently came across a theorem which makes a hypothesis that a certain measure space is separable (the definition can be found here). In order to avoid confusion, I'll add the definition here: We ...
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Noob Question : Need help to understand : Probability with Martingales : page 24

Coin toss infinitely often. The sample space is $\Omega = \{ H , T \}^{\mathbb{N}} $ And we do not have any problem. And a typical point is $\omega = \omega_1 \omega_2 \omega_3 ... $ where $\omega_n ...
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Is every sigma-algebra generated by some topology?

Well, my question is precisely what the title says: Is every sigma-algebra on a set $X$ generated by some topology on $X$? Actually,I am unable to either prove or create a counterexample, but I have a ...
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Noob Question : Need help to understand : Probability with Martingales : page 25

I was asked to post the question from here I am trying to learn measure theoretic probability. The book I am trying to learn it from is Probability With Martingales and, I am really not understanding ...
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How a 1-point set can have positive probability measure?

Suppose I have a program: x := bernoulli() if (x == true): return 0.5 else: return uniform-continuous(0,1) If I am not mistaken, the distribution out output ...
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If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too?

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too? My outline is the following: Since $X,Y$ is a measure zero set, ...
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66 views

What does it mean to sample, in measure theoretic terms?

Suppose I have some random variable $X$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What does it mean, in measure theoretic terms, to draw a sample from $X$? When $\Omega$ ...
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1answer
27 views

Change of variable in Hardy Littlewood proof

This is part of a proof I try to understand. Lets $Tf(x)$ be the Hardy littlewood maximal funtion, $$Tf(x) = \sup_{r>0} \frac{1}{B(r,x)} \int_{B(r,x)} f(y) dy$$ and $E_\lambda = \{y: |Tf(y) |> ...
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104 views

Is a sigma algebra (up to null sets, as in conditional expectation) always generated by a random variable?

Motivation Let $(\Omega, \mathcal F, \mathbb P)$ be a standard probability space. For two $\mathcal H, \mathcal G \subset \mathcal F$, we say $\mathcal H = \mathcal G$ mod 0, if they are same up to ...
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44 views

Why is any Lebesgue-Stieltjes measure on $(\mathbb{R},\mathcal{B}_\mathbb{R})$ not complete?

Let $F: \mathbb{R} \to \mathbb{R}$ be any increasing, right-continuous function. Then we define the Lebesgue-Stieltjes measure associated to $F$ to be the unique measure $\mu_F$ on ...
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62 views

Hardy-Littlewood maximal theorem (Marcinkiewicz)

I have two pages from a book called "Garnett" and I will present Hardy-Littlewood maximal theorem in class on Wednessday. The theorem is stated: if $f\in L^p(\mathbb{R}), 1 \leq p \leq \infty,$ then ...
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28 views

Lebesgue measure of symmetric difference [duplicate]

I am completely lost with this problem. I was wondering if anyone had ever seen it before and/or had any ideas on how to solve it. Take $\bar{\lambda}$ as the measure on ...
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1answer
40 views

how many borel subsets of reals are there?

Can someone provide me the proof, or atleast a link to where I can find the proof of the fact: there are as many borel subsets of the reals, as there are real numbers? Of course, I am assuming AC.
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3answers
47 views

Show that $f=g$ when $\int_0^\varepsilon f dx = \int_0^\varepsilon g dx$ for all $\varepsilon\in\mathbb{R}$

I know that this should be pretty straightforward to prove, but I'm trying to find a fairly clean proof for this. I have one that seems to be far too complicated, and I'm wondering if this is ...
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113 views

Lebesgue measure of symmetric difference with translation.

The problem is the following Let $E\subseteq\mathbb{R}$ be a Lebesgue measurable set such that $\lambda(E\Delta (E+x))=0$ for every $x$ in a dense subset of $\mathbb{R}$. Then either ...
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Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
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Markov Processes: How to show $\int P(X_t\in B\mid X_0)dPX_0^{-1}=P(X_t\in B)$?

Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the ...
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42 views

Trace of a measure on a subset (or restriction of a measure to a subset)

This is an exercise from Measure Theory by Cohn. Given a measurable space $(X,\mathcal{A})$ and subset $C$ which may not be measurable, we can form the trace of $\mathcal{A}$ on $C$ denoted ...
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When do invariant measures arise from smooth differntial forms?

It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq ...
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A problem on Lebesgue dominated convergence theorem

I have the following 2 problems for homework, and I couldn't do the 1st one and need to check if my solution is correct for the 2nd one thanks 1) If $f$ is an integrable function on $ \Bbb R $ such ...
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Measurability of the set of all continuous functions

Consider $\mathbb{R}^\mathbb{R}$ the set of all real-valued functions of real variable with the product topology. This topology induces the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^\mathbb{R})$. ...
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195 views

If two measures agree on generating sets, do they agree on all measurable sets?

Here's the problem that insprired my question: Suppose $X$ is the set of real numbers, $\mathcal B$ is the Borel $\sigma$-algebra, and $m$ and $n$ are two measures on $(X, \mathcal B)$ such that ...
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49 views

$\sigma$-algebra and measurable set.

Today I am reading David Williams's Probability with Martingales. In chapter one, He introduce the notion of Measurable space: A pair $(S,\Sigma)$,where $S$ is a set and $\Sigma$ is a ...
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1answer
46 views

If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?

Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad ...
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42 views

Equality of measures, symmetric intervals, compact, bounded, measurable sets

Let $\mu$ be a measure on $L_m$ , ($m \ge 1$) - $\sigma$ - algebra of Lebesgue-measurable sets, such that its values on compact symmetric intervals (cubes) are equal to Lebesgue measure of those ...
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1answer
336 views

Gaussian Matrix Integral

I need your help to solve this exercise : Let $S$ be a symmetric Hermitian matrix $N\times N$ : $S=(s_{ij})$ with $s_{ij}=s_{ji}$. When $\langle s_{ij}s_{kl}\rangle\neq 0$ What $$\int ...
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154 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
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53 views

Countable sum of atomic measures is atomic?

Let $(X,\Sigma)$ be a measurable space and $(\mu_n)$ a sequence of atomic measures defined on this space. Recall that a measure $\mu$ is atomic if for any measurable $A$ of measure $\mu(A)>0$ there ...
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1answer
30 views

For a probability kernel $\mu$, why is $\left\{ (s,t):\mu(s,[0,x])<t \right\}$ a measurable set?

Let $\mu$ be a probability kernel from a measurable space $(S,\mathscr S)$ to $([0,1],\mathscr B[0,1])$. Let $s\in S, t\in [0,1]$. I don't understand why the function$$f(s,t)=\sup \left\{x\in ...
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65 views

Computing Newtonian capacity of sets like intervals, discs?

For a metric space $(E,\rho)$ the $a$-capacity is defined as $$\mathrm{Cap}_{a}(E)=\left[\inf\left\{\int \int \frac{d\mu(x) \, d\mu(y)}{\rho(x,y)^{a}}:\mu\text{ probability measures on ...
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How to show that this function is measurable.

For a given $N$, I have this function: $0$ when $0 \le x \le \frac{1}{N+1}$ $\sin\left(\frac{1}{x}\right)$ when $\frac{1}{N+1}<x\le\frac{1}{N}$ $0$ when $\frac{1}{N}<x\le 1$ Here is my ...
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1answer
66 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
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Limit a.s. of a sequence of normal random variables is normal.

I know that the statement "If $X_n$ is a sequence of normal random variables which converges a.s. to a random variable $X$, then $X$ is also a normal random variable" is true. However, do you ...
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188 views

What is the difference between $\sigma$-algebra, $\sigma$-ring, and field of sets?

I don't really understand the difference between these stuff. They look really similar. What is the difference between those? Which one do people use in measure theory, and probability related things ...
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Subadditivity of outer measure on real numbers [duplicate]

In the field of measure theory, the outer measure is defined on the set of real numbers as: $$m^{*}(A) = \inf\left\{ \sum_{n=1}^{\infty} l(E_n): \space A \subset \bigcup_{n=1}^{\infty} E_n \right\} ...
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66 views

Outer measure is countably subadditive

Working on the proof that outer measure is countably subadditive in Royden For a set $A \subset X$ we define the outer measure: $$\mu^{*}(A) = \inf\left\{ \sum_{n=1}^{\infty} \tau(T_n): \space T_n ...
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1answer
35 views

Semialgebra logical error

I was following a proof on Rosenthal's, A first look at rigorous probability theory book, but I believe that one step flawed. Let me set up: $\mathcal{J}$ is a semialgebra of subsets of $\Omega$. ...
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768 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
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33 views

Bounded Linear Transformation proof

One paragraph in my text is to prove that $\|T\|=\sup\{|\langle Tf, g\rangle|:\|f\|<1, \|g\|<1\}$, where we have a bounded linear operator between two Hilbert spaces $T:\mathcal H_1\rightarrow ...
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can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
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93 views

The walk of a knife

"A knife is slowly moved parallel to itself over the top of a cake. At each instant the knife is poised so that it could cut a unique slice of the cake. As time goes by the potential slice increases ...
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1answer
120 views

Lebesgue measure compact symmetric intervals

Let $\mu$ be a measure on $L_m$ , ($m \ge 1$) - $\sigma$ - algebra of Lebesgue-measurable sets, such that its values on compact, symmetric intervals (cubes) are equal to Lebesgue measure of those ...
3
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2answers
42 views

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as ...
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176 views

Is every continuous function measurable?

In non-Hausdorff topology it is standard to define the Borel algebra of a topological space $X$ as the $\sigma$-algebra generated by the open subsets and the compact saturated subsets. Recall that a ...
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29 views

outer measure for definition question

I'm reading Pugh's "Real Mathematical Analysis" and just started the chapter on Lebesgue Theory. I guess you'd have to a copy available to answer this question, but on page 374, he defines the outer ...
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Understanding Borel-Cantelli Lemma

I am having some difficulty in understanding the importance of this lemma; Let {$E_k$}$_{k=1}^{\infty} $be a countable collection of measurable sets for which$ ∑_{k=1}^∞ m(E_k)<∞$. Then almost all ...
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59 views

A question on Lebesgue Measure

I am trying to show if $E_k \subset (0, 1)$, $(k=1,...,n)$, and $\sum_{1 \le k \le n}{\mu(E_k)} > n -1$, then $\mu(\bigcap_{1 \le k \le n}{E_k}) > 0$. Intuitively this seems like such an ...
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79 views

Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
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1answer
47 views

Application of weak $L^p$ estimate besides for proving boundedness of some linear operator

For all $1\leq p< \infty$, weak-$L^p(\mathbb{R}^d)$ space is defined as a set of all functions $f$ such that $$\gamma^p|\{x\in \mathbb{R}^d: |f(x)|>\gamma\}|<\infty$$ for every ...