Questions tagged [measure-theory]

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

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measure of the set $ |G(x_1, x_2, \dots, x_n)| \leq \lambda $

If $G$ is a homogeneous polynomial of degree $d$ in $n$ variables, then what is the measure of the set of points satisfying $$|G(x_1, x_2, \dots, x_n)| \leq \lambda $$ A reference suggests that it is ...
zero2infinity's user avatar
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Showing that an Algebra is not a $\sigma$-Algebra

In a script I found the following example for an algebra, which is not a $\sigma$-algebra: Let $$\mathcal{I} = \{ \emptyset, \mathbb{R} \} \cup \{ (a, b] : -\infty < a < b < \infty \} \cup \{ ...
SineOfTheTimes's user avatar
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New variable with same density under change of measure

If $X$ is a random variable defined on $\mathbb{R}^n$ endowed with the Borel sigma-algebra and the Lebesgue measure $\lambda$. X has a density, $f$. For $k>0$, consider the density $$g(x_1,\ldots,...
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How to understand that a subset of a manifold is Lebesgue measurable?

I have recently been reading Werner Ballmann's Introduction to Geometry and Topology, and when I get to section 7 of chapter 3 (where the author wants to define the oriented integral), he says "...
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If $f$ is measurable and $f=g$ almost everywhere is $g$ measurable

We were told in lecture, that if two functions $f,g$ are equal almost everywhere. Meaning that for a measure space $(X,A, \mu)$ sucht that the set $\{x \in X | g(x) \neq f(x)\}$ is a measurable ...
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About the definition of Radon measure on Kallenberg's book

I believe there are several definitions for Radon measure, which are equivalent under certain conditions. The one in the Wikipedia says that a measure $m$ on the $\sigma$-algebra of Borel sets of ...
ElectronicKid's user avatar
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X,Y,Z are independent identically distributed continuous then $X > Y \cap X > Z$ and $Y > Z$ are independent

I'm trying to prove that If $X,Y,Z$ are independent identically distributed continuous random variables then the events $X > Y \cap X > Z$ and $Y > Z$ are independent. I start with proving $...
Ta Thanh Dinh's user avatar
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Concentration-compactness Lemma

Let $N\geq 3$ y $2^{*} := 2N/(N - 2)$. The space $\mathcal{D}^{1,2}(\mathbb{R}^N) := \left\lbrace u \in L^{2^{*}} (\mathbb{R}^N) : \nabla u\in L^2(\mathbb{R}^N)\right\rbrace $, with the inner product $...
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Adapted Stochastic Process intuitions in previous answers seem misleading

Set up Given a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_t\}_{t\in T})$ people seem to say that the intuitive meaning of the stochastic process $\{X_t\,:\, t\in T\}$ ...
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Is $\bar{\mathcal{E}}=\mathcal{E}\cup F$, where $\mathcal{E}$ is the set of all elementary subsets of $\mathbb{R}$ and $F$ is finite or countable?

Let $m^*$ be an outer measure and $\Delta$ denote the symmetric difference. Define a semi-metric $d: \mathscr{P}(\mathbb{R}) \times \mathscr{P}(\mathbb{R})\to [0, \infty]$ by $d(A, B)=m^* (A \Delta B)...
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Write $\mathbb{P}^X(A)$ as $\mathbb{E}[1_A X]$

$(\Omega, \mathcal{F}, \mathbb{P})$ probability space $(E, \mathcal{B}(E))$ a measurable space where $B(E)$ is the Borel sigma algebra and $E$ is a Banach space $X:\Omega\to E$ random variable with ...
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When is the composition of Markov kernels well-defined?

Wikipedia, and all resources I checked, seem to suggest that we can always compose two Markov kernels. Let $(\mathsf{X}, \mathcal{X})$, $\mathsf{Y}, \mathcal{Y})$, $(\mathsf{Z}, \mathcal{Z})$ be ...
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I proved any linear subspace of $\mathbb{R}^n$ is closed in $\mathbb{R}^n$ to prove $\overline{T}$ has measure zero. (Munkres "Analysis on Manifolds")

I am reading "Analysis on Manifolds" by James R. Munkres. Theorem 20.1. Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation $h(x)=A\cdot x$. ...
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A proof of Polya-Szegö inequality

Denote by $|A|$ the $N-$dimensional Lebesgue measure of a Borel set $A \subset \mathbb{R}^N$ and define $$ A^\ast := B_{R}(0), \quad R = \left(\frac{N}{\omega_N}|A| \right)^{\frac{1}{N}}, $$ where $\...
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A homework problem from Bruce Driver's book "Probability Tools with Examples"

Below is the problem from Exercise 5.12 in Bruce Driver's book "Probability Tools with Examples". I was wondering how to solve it. Let $F, G:[0,1] \rightarrow \mathbb{R}$ be two non-...
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Particular version of disintegration theorem

I'm looking for a published reference for a proof of the following theorem, a version of the disintegration theorem. Let $(X,\mathcal{A})$ be a standard Borel space and let $\mu$ be a probability ...
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How to compare $\infty$ with another $\infty$ in the proof of sub-additivity property.

When reading a proof of the sub-additivity property of the outer measure on $\mathbb{R}$, that is: If $\lbrace A_n \rbrace_{n=1}$ in $\mathscr{P}(\mathbb{R})$ then $$ m^*\left( \bigcup_{n=1}^{\infty}\ ...
Tran Khanh's user avatar
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Counter examples of $|m^*(A)-m^*(B)|\le d(A, B)$ if $m^*(B)=\infty$, where $m^*$ is an outer measure and $d$ is a semi-metric

Let $m^*$ be an outer measure on $\mathbb{R}$, i.e. for $A\subseteq\mathbb{R}$ then: $$ m^*(A):=\inf\left\{\sum_{n=1}^{\infty}\mathscr{l}(I_n): I_n\text{ is an open interval and } A\subseteq \bigcup_{...
Tran Khanh's user avatar
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Lebesgue Measure of Closed and Open Sets

I am given the definition of the Lebesgue Measure: Lebesgue Measure $\mathcal P$ on the Borel $\sigma$-algebra $\mathcal B[0,1]$: $\mathcal P[[a,b]] = b - a$ for all $0 \leq a \leq b \leq 1$ The ...
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Question concerning the correctness of this version of Fatou's Lemma

In lecture we learned about Fatou's Lemma stated as follows: Let $(X, \mathcal{S}, \mu)$ be a measure space and $(f_k : X \to [0,\infty])$ measurable and $f: X \to [0, \infty] $ a function such that: $...
user007's user avatar
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Increasing limit in sequence of set [closed]

I am reading Foundations of Modern Probability by Kallenberg and stuck on the meaning of increasing limit $A_n\uparrow A$ for sequence of set (pageno 1), I read this question, but I have to confirm ...
zia badar's user avatar
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Extension of Borel map from a separable metric space to a Polish space

Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$. Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an ...
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prove that if $A \cap B = \emptyset$ and $m^*(A \cup B) \neq m^*(A) + m^*(B)$

The following exercise analyzes the additivity of $m^*$. a) Prove that if $A$ is a measurable subset of $\mathbb{R}$, then for every $B \subseteq \mathbb{R}$, it holds that $$m^*(A \cup B) = m^*(A) + ...
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Is it possible to fill out $n$ dimensional cube with open disjoint disks, such that the difference is a lebesgue zero measure? [closed]

Let $s, \delta \in \mathbb{R}$, $x \in \mathbb{R}^n$ and $W = \prod_{i = 1}^n [x_i - s, x_i + s]$. Do there exist pairwise disjoint open disks $(K_i)_{i \in \mathbb{N}}$ such that $K_i \subseteq W^{°}...
baleine6's user avatar
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Graph of a measurable function is measurable (not on metric spaces)

Im trying to solve the following problem: Problem Let $(S, \mathcal{A}) , (S, \mathcal{A} ')$ be measurable spaces and $\varphi, \psi , \psi' : S \to S' $ $\mathcal{A}- \mathcal{A}'$ measurable ...
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Is it possible to define $L^p$ spaces using a non-sigma-finite measure space and a Banach space?

Most often (at least in probability), one defines the $L^p$ space as Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $p\geq 1$ be a real number. Then $$ L^p(\Omega, \mathcal{F},...
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Defining the expectation of a measurable function with respect to a (non-probability) measure

The typical definition of expectation requires a probability space and a random variable Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable ...
Euler_Salter's user avatar
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2 votes
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Equivalent condition for measurable filters.

It's a known fact (see for example Bartoszynski and Judah: Set Theory, On the Structure of the real Line) that for a filter $F$ on $\omega$ the following conditions are equivalent: $F$ is Lebesgue ...
Caro Meier's user avatar
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proof that the extended real number is equal to outer measure

For a subset $A \subseteq \mathbb{R}$, the extended real number is defined as $$ N(A) = \inf \left\{ \sum_{k=1}^{\infty} \ell(I_k) : A \subseteq \bigcup_{K=1}^{\infty} I_k\right\}$$ where the $I_k$ ...
satl's user avatar
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Laplace Transform of a Piece-wise function with a Weibull distribution.

Suppose I have the following piecewise function: $$Q(t) = \begin{cases} W(t) & t<T \\ 1 & t=T \\ 0 & t>T \end{cases}$$ ...
Keyvan's user avatar
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Example of finding the lower semi-continuous convex function whose subdifferential contains the support of an optimal transport

Related to this post, but slightly altered. Let $\mu$ uniform on $\{0\}\times [0,1]$, $\nu$ uniform on $\{-1,1\}\times [0,1]$. Consider the Kantorovich optimal transport problem of $\mu$ to $\nu$ with ...
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Am I understanding right about the sigma algebra generated by random variables.

I have encountered many types of sigma algebra generated by a random variable. I would like to clarify my understanding of them. The simplest one is generated by a random variable. I skip it here. ...
Andrew_Ren's user avatar
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1 answer
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Determine all $(p,q)$ where Lorentz quasi-norm $\|\cdot\|_{L^{p,q}}$ is norm

On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\cdot\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
Liding Yao's user avatar
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Prove that $\mathcal{F}=\bigcup^\infty_{n=1}\sigma(\{\{1\},..\{n\}\})$ is a field but not a $\sigma$-field on $\mathbb{N}$

Consider the sigma fields on $\mathbb{N}$: $F_n=\sigma(A_n)$, where $A_n=\{\{1\},\{2\},\ldots,\{n\}\}$ for $n\ge1$. Define $\mathcal{F}:=\bigcup^\infty_{n=1}F_n$. Prove that $\mathcal{F}$ is a field ...
reyna's user avatar
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Let $p$ be a probability dist. on $X\times Y$ s.t. marginal of $p=\mu$. Is $p\otimes p(A\times (Y\times Y)) = \mu\otimes \mu(A)$ true?

Let $X$ and $Y$ be complete and separable metric spaces with Borel probability measures $\mu$ and $\nu$, respectively. Now take any distribution $p$ on $X\times Y$ so that the marginal of $p$ on $X$ ...
Kaira's user avatar
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Show that there exists a sequence of continuous functions $\{g_n\}$ so that $g_n\to f$ as $n\to \infty$

Following this question:Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions. Let $f\in L^1(\mathbb{R})$. Show that ...
Hermi's user avatar
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Confusion on Folland's redefinition of L^1 space.

The following are excerpts from Folland: Proposition 2.12: 2.12 Proposition. Let $(X, \mathcal{M}, \mu)$ be a measure space and let $(X, \bar{\mathcal{M}}, \bar{\mu})$ be its completion. If $f$ is an ...
juekai's user avatar
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What topology $\mathcal{T}$ on $\mathbb{R}$ generates the Borel sigma-algebra on $\mathbb{R}$? [closed]

Given a topological space $(\mathsf{X}, \mathcal{T})$ where $\mathsf{X}$ is a set and $\mathcal{T}$ is a topology on it, one can generate a sigma-algebra using this topology: the Borel sigma algebra $$...
Physics_Student's user avatar
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1 answer
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Intersection of uncountably many sigma algebras

The intersection of sigma algebras is a sigma algebra $$ \bigcap_{\alpha\in \mathcal{A}} \mathcal{X}_\alpha $$ where $\mathcal{A}$ is an index set. However, in every definition I looked at (...
Physics_Student's user avatar
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Explanation related outer measure and measurable cover

Theorem C is from the book Measure Theory by Halmos...from Chapter 3. My question: how Sigma finite of E is used in the proof? It will be very much helpful kindly give some hint or explanation.
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Exercise 2.14 in Villani's "Topics in Optimal Transportation"

Exercise 2.14 in Villani's "Topics in Optimal Transportation" asks us to show the following: Let $Q=[0,1]^{n-1}$, and $\mu$ is uniform on $Q\times \{0\}$, $\nu$ uniform on $Q\times \{1\} \...
uniform_on_compacts's user avatar
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If $X$ is a centered random variable and $C$ its covariance matrix then $P(X \in \mathrm{Im } C) = 1$

Let $X$ be an $m$-dimensional random variable and let $C$ be its covariance matrix. I am following a proof that says if $X$ is centered then $P(X \in \mathrm{Im } C) = 1$ where $\mathrm{Im } C$ is the ...
CBBAM's user avatar
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Showing that $T$ is Hilbert-Schmidt operator

Suppose $H = L_2(B)$ where $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B \times B$ that satisfies $|K(x,y)| \leq A|x-y|^{-d+\alpha}$ for some $\alpha > 0$ ...
Grigor Hakobyan's user avatar
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Total variation convergence in context of stochastic processes

Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution $$X_t~\...
Oskar Vavtar's user avatar
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Convergence of the sequence of the pushforward measures of weakly convergent sequence [closed]

Let $S$ be a Polish space endowed with its Borel $\sigma$-field $\mathcal{B}(S)$, and let $(A,\mathcal{A})$ a measurable space. Let $(\mu_n)$ a sequence of probability measure on $(S,\mathcal{B}(S))$ ...
Abdessamad Jawad's user avatar
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Understanding the term almost everywhere in measure theory

I am trying to understand the term "almost everywhere" from measure theory correctly. So given two extended real-valued integrable functions $f, g: X \rightarrow \bar{\mathbb{R}}$ with $$\...
guest1's user avatar
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Measurable function and subspace sigma algebra [closed]

Let $(X, \mathcal{A})$ and $(Y, \mathcal{F})$ be two measure spaces and $f:X\rightarrow Y$ a function. 1.) If we assume that the image of $f$, $D:= Image(f)$ is a true subset of $Y$, and we further ...
guest1's user avatar
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2 answers
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What does "convex class of probability measures" mean in the definition of scoring rules?

Taken from Wikipedia (here), a scoring rule has the following definition Let $\Omega$ be a sample space, and $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$. Let $\mathcal{P}$ be a convex ...
Vicky's user avatar
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1 answer
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A set which only exists in the $\sigma$-algebra completion

Say I have a $\sigma$-algebra $\mathcal{F}$ and its completion $\mathcal{F}^{\ast}$. If a set $E\in \mathcal{F}^{\ast}$ but $E\notin \mathcal{F}$, does that mean that I definitely have sets $A,B\in\...
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Measure decaying faster than exponential

I am following Glimm & Jaffe's Quantum Physics where they define a generating functional $$S\{f\} = \int e^{i\phi(f)} d\mu.$$ Here $d\mu$ is a Borel probability measure over the space of ...
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