Questions tagged [measure-theory]

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

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Jordan decomposition of $F \in BV$

Suppose $F \in BV$, then its Jordan decomposition is given by $F = \frac{1}{2}(T_F + F) - (T_F - F)$ where $T_F$ is the total variation of $F$. In Folland's text he says since $x^+ = \max(x, 0) = \...
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Lemma on measures in Banach spaces

I've been reading some semi-group theory books and handbooks, and I came across this lemma (below) that i used later on in a handbook to show a theorem. My problem is I haven't found any proof to it ...
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Prove that $\nu$ is absolutely continuous w.r.t. $\mu$ iff $\sum \alpha_j^2<\infty$

This is question 3 in Chapter 4.12 from Barry Simon - Real analysis I tried to define $ f_j : \{0,1\} \to \mathbb{R} $ where $n$ means that it is a function from $j$th $\{0,1\}$ in the product to $\...
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Weak Convergence for a continuous function without compact support

Let $X$ be an open subset of $\mathbb R^n$ and let $\Omega$ be a relatively compact, open subset of $X$. Let $\{ \mu_n\}$ be a sequence of positive measures that converge weakly to the positive ...
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Complex valued Stieltjes integrals : If $f\in\mathcal{R}(\alpha)$, do we have $\mathfrak{Re}(f), \mathfrak{Im}(f)\in\mathcal{R}(\alpha)$?

Given a bounded complex function $\alpha:[a,b]\to\mathbb{C}$, we can define the Riemann-Stieltjes integral of $f:[a,b]\to\mathbb{C}$ (also bounded) in a way that is very much analogous to the usual ...
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Show that if $\{x: f(x)\neq g(x)\}, \{x: g(x)\neq h(x)\}\in\mathfrak{M}$ then $\{x: f(x)\neq h(x)\}\in\mathfrak{M}$.

The Problem: Show that if $\{x: f(x)\neq g(x)\}, \{x: g(x)\neq h(x)\}\in\mathfrak{M}$, then $\{x: f(x)\neq h(x)\}\in\mathfrak{M}$, where $f, g, h$ are measurable functions. The problem arises from ...
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approximated continuity

what is an example of measurable function $f$ :$\mathbb R^n$ $\to$ $\mathbb R$ with a point $x_0$ that is not a point of approximated continuity?
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Is this function arising from measure of a certain set continuous?

Let $F$ be a CDF of random variable $X$ distributed in $[0,1]$. $F$ has a density, $f$. Let $\mu_F$ denote the corresponding measure, i.e. $\mu_F(A)=\int\limits_{A}f(x)dx$ for all $X \in \mathcal{B}([...
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Is the restriction of $\sigma(\mathcal E)$ to a subset $U$ equal to $\sigma(\mathcal E\cap U)$?

Suppose $\mathcal A=\sigma(\mathcal E)$ is a $\sigma$-algebra on $X$ and $U\subset X$ is some subset. We know that $\mathcal A_U=\{A\cap U:A\in\mathcal A\}$ is a $\sigma$-algebra on $U$, but do we ...
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estimate of measure of a $C^1$ range set

Let $\Omega\subset\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ be a $C^1$ function . Denote $J_f(x)=\det(f'(x))$ . Prove that $$\mu(f(X))\leq\int_X|J_f(x)|dx$$ for any $X\subset\Omega$ ...
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Conceptual problems in measure theory.

I am studying measure theory as a graduate student. I have studied the theory up to integration. Now I want to test my concepts by doing some problems as the best way to learn a topic is to do ...
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Is there a relation between s-finite and semifinite measures?

I just encountered the concept of semifinite measure (apparently also called locally finite measure or measure with the finite subset property), defined as follows: A measure $\mu$ is called ...
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On the equivalence of two definitions of null sets

I am currently reading the lecture notes of two different professors and I would like to verify/understand that their definitions of null sets are equivalent. Let $(X,\Sigma,\mu)$ be a measure space ...
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Showing that two random variables are equal in distributions [closed]

I have to show that the given random variable is equal in distribution to the expression on the right-hand side. For reference, the given process we are studying is the solution to the Ornstein-...
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$\sigma$-algebra generated by uncountable random variables

Let $(X_1,\mathcal{A}_1)$, $(X_2,\mathcal{A}_2)$, and $(X_3,\mathcal{A}_3)$ denote three measurable spaces. Let $\mathcal{G}$ denote a set of (pontentially uncountable) measurable functions $g:X_2\to ...
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For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X$, and a function $g$, is $g(X)$ automatically measurable?

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X: \Omega \to D$, and an arbitrary function $g: D \to E$, is $g \circ X$ also measurable and thus a random variable? I ...
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Approximating $\mathcal X \subset \mathbb R^n$ with the union of disjoint hyperrectangles

Let $\mathcal X$ be a bounded subset of $R^n$ that is generated by a set of linear inequalities. For example, let ${\bf x} = (x_1, x_2, x_3, x_4, x_5)^\intercal \in \mathcal X$ iff \begin{align*} 0 &...
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Defining a function without using Axiom of Choice

I have a situation where I do not know if I need the axiom of choice: Let $\mathcal{B}(\mathbb{R})$ be the collection of Borel measurable subsets of $\mathbb{R}$. I have a (possibly non-Borel) subset $...
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$\sup\{\lambda_d(A): A \subseteq \mathbb R^d \text{ at most countable}\}$

How can I calculate $\sup\{\lambda_d(A): A \subseteq \mathbb R^d \text{ at most countable}\}$ I know since $A$ is countable, I can write $A$ as a union of its elements.
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Ergodicity on a finite set

Let $\Omega$ be a finite set ($ \#\Omega = n$), how many dynamical systems on $\Omega$ are ergodic?
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If the set $f^{-1}\{a\}$ is measurable for all $a$, then is $f$ a measurable function?

What can be said about the measurability of an extended real valued function, defined on $\mathbb{R}$ such that the set $\{x:~f(x)=a\}$ is measurable for each extended real value $a$? I think that $f$ ...
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Total variation of $F:[a,b]\to\mathbb{R}^{N}$ where each component of $F$ is given by a definite integral

Let $\alpha:[a,b]\to\mathbb{R}$ be monotone increasing and let $\mathcal{R}(\alpha)$ be the set of all functions $f:[a,b]\to\mathbb{R}$ that are Riemann-Stieltjes integrable with respect to $\alpha$. ...
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Under what minimum assumptions a null set is necessarily meager?

$A\subset \Bbb{R}$ is a null set if $\lambda(A) =0$ ($\lambda$ :Lebesgue measure) $A\subset \Bbb{R}$ meager if $A$ is countable union of nowhere dense sets (sets whose closure contains no nonempty ...
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1 vote
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Extending a function that gives a value to convex functions to a measure

I am wondering if such a result exists (or similar) and or if there is a "simple" proof. Let $\mathcal X$ be a bounded and closed subset of a topological vector space, let $\Sigma$ be the ...
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2 votes
2 answers
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Deduce $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}a_{ij}$ using Lebesgue's Convergence Theorem.

The Problem: Given $a_{ij}\geq0$ for $i, j=1, 2, 3,\dots$, deduce $$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}a_{ij}$$ using Lebesgue's Convergence Theorem. ...
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Counting measure has no Lebesgue decomposition proof verification

I am working through exercises in Rudin RCA but I have some questions about whether my justification is valid as it differs from other posts on the site. Let $\mu$ be the Lebesgue measure on $(0,1)$ ...
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Müntz-Szasz theorem: assumption measure is concentrated in (0,1]

I'm studying the proof of the Müntz-Szasz theorem of Rudin's book.. They define the function $$f(z)= \int_{I} t^z d\mu(t)$$ and we may assume that the measure is concentrated in (0,1]. But why is this ...
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3 votes
1 answer
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How to find a fractal with a predetermined Hausdorff dimension? [duplicate]

For many patterns that display self-similarity, the Hausdorff dimension can be found. Sometimes the dimension is calculated and approximate - as is the case with the Feigenbaum attractor - but often ...
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When does the asymptotic density of a set in the natural numbers not exist?

Let $\displaystyle d(A)=\lim_{n\to\infty}\frac{\# (A\cap[1,n])}n$ be the asymptotic density of $A$ in the natural numbers. I came across a statement saying that such that the limit exists it measures ...
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1 vote
1 answer
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Measure of image of an absolutely continuous function and its integrals

Consider an absolutely continuous function $F:[a,b]\rightarrow\mathbb{R}$. One can find a function $f\in L^1([a,b],m)$, where $m$ is a Lebesgue measure, such that $F(x)-F(a)=\int_a^xf(t)dt$. I am ...
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Is there a proof that the power set of a countable sample space supports a probability measure?

When I typed my question the following link came up in the 'similar questions' list and I thought great, there's an answer to my question. Why can a probability measure be defined over power set of ...
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1 answer
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Proof of convergence of random variables in $L^p$ via convergence in probability and uniform integrability.

Consider the following proposition. Part (i) i have no problem with. Its the proof of part (ii) that (because of my lack of knowledge of advanced measure theory) am having trouble in understanding. ...
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4 votes
0 answers
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Can we find $f\in \Bbb{R}^{[0, 1]}$ with the property $\mathcal{M}$ which doesn't satisfy the property $\mathcal{B}$?

$f:[0, 1]\to \Bbb{R}$ be a function. $f$ satisfy the property $\mathcal{M}$ of $f(A) $ is meagre for every $A\subset [0, 1]$ meagre. $f$ satisfy the property $\mathcal{B}$ if $f(A) $ is a set with ...
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2 votes
1 answer
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Preimage of measure 0 set under norm function has measure 0

Let $A \subseteq [0,1]$ be a null set with respect to the Lebesgue measure. Is it true that $\{x\in \mathbb R ^k: |x| \in A\}$ is a null set in $\mathbb R^k$, for all natural $k$ ($|(x_1,\dots,x_k)| = ...
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If $\mu_i \to \mu$ weakly, then do we have $\operatorname{supp}(\mu_i) \to \operatorname{supp}(\mu)$ in Hausdorff distance?

Suppose $\mu_i$ are Radon measures with support in $\mathbb{R}^{n}$. Is it true that if $\mu_i \to \mu$ weakly then their supports must converge in Hausdorff distance?
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What functions are continuous in the Wasserstein metric?

Consider the space $L_p(\mathbb{R})$ of probability measures on $\mathbb{R}$ with $p$th moments. This set is a metric space under the $p$-Wasserstein metric $$W_p(\mu,\nu)^p=\inf_{Z=(X,Y):X\sim\mu,Y\...
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1 vote
1 answer
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Definition of essentially T-invariant function

A function is $T$-invariant if $f(T(x))=f(x)$ for all $x\in X$. In text book: Introduction to Dynamical system by Brin, it defines essentially $T$-invariant: if $f(T(x))=f(x)$ almost every for $x\in X$...
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2 votes
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Abstract integral interpretation

I have been studying measure theory and it has occurred to me that so far I have not developed the intuition behind an abstract integral. I believe I understand what it is in technical terms, but I'm ...
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2 votes
0 answers
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Limiting probability measure on an increasing sequence of finite products

$\newcommand{\mc}{\mathcal}$ Let $(X_i, \mc X_i)$ be a measurable space for $i=1, 2, 3, \ldots$. Let $X^{(i)}$ denote the set $X_i\times X_{i-1}\times \cdots \times X_1$ and we equip it with the ...
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1 vote
0 answers
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Approximating an open set in measure with another open set whose boundary has zero measure

Given a bounded open set $D\subseteq\mathbb R^n$. I'm wondering whether it is always possible to find another open set $A\subseteq D$ such that the boundary of $A$ has zero Lebesgue measure and $D-A$ ...
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1 vote
1 answer
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+50

Exemple where tower property of conditional expectation is NOT verify

Question: Let $\Omega=\{a,b,c\}$. Give an example for $X, F_1, F_2$ in which $E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$ My answer: I am not at all sure of my answer. If you have any shorter and nicer ...
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11 votes
3 answers
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Prove $\sum_{i,j:i<j} μ(E_{i} \cap E_{j} )=\infty$.

The problem is: Assume that $(X,\mathcal{A} , μ)$ is a finite measure space (i.e., $μ(X) < \infty $ ) and the sequence of sets $E_{j}\in \mathcal{A}$ satisfies $\sum^\infty_{j=1} μ(E_{j} )=\infty$....
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3 votes
2 answers
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Does the set of convex combination of points in Cantor set contains a non empty open interval?

$\mathcal{C}$ denote the cantor middle third set. $$\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$$ $\mathcal{C}_0=\mathcal{C}_1=\mathcal{C}$ and we can prove that that $\mathcal{C}$ contains no ...
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0 votes
1 answer
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Complement of the support of measure on R^n has measure 0

How do we prove that $\mu (B) = 0$ for any ball $B$ such that $B \subset S^c$? For any $x \in S^c$, there exists $r_x > 0$ such that $\mu \left( B(x, r_x) \right) = 0$. Why there exists a countable ...
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1 vote
1 answer
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Union of spheres is a null set

It is proven that the sphere $S(0^{[k]};a)$ is a null set in $R^k$ for every $a \in R$. Question: Let $C \subseteq R$ be a null set. Prove that $\bigcup_{a \in C} S(0^{[k]};a)$ is a null set in $R^k$. ...
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0 votes
1 answer
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Preimage of a sufficiently small set

I think the following claim is correct but I don't know how to prove it rigorously: Assume $A_n\subset \mathbb{R}$ a sequence of measurable sets with $m(A_n)<\epsilon$ for sufficiently large $n$. $...
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  • 301
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1 answer
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Egorov theorem Stein book page 33

The proof defines$$E_k^n=\left \{x\in E:|f_j(x)-f(x)|<\frac{1}{n}\forall j>k\right \}.$$Fixing $n$ and letting $k\to \infty$ we have then $E_k^n\to E$. So for some $k_n$ we have $m\left (E-E_{...
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1 vote
1 answer
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Lebesgue integration theory

So my question is, we define the simple function to take finite values, does it change a lot of what we have established for Lebesgue integration theory if we define the simple function to take ...
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-2 votes
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Does pointwise convergence imply measurability? and why?

$E$ is a Banach space and for every $u_0 \in E$ and $\{u_n\} \subset E$ with $u_n \to u_0$, we have $$S(t)u_n \to S(t)u_0$$ pointwise in $t \geq 0$. Why $t \mapsto S(t)u_0$ is measurable there ?
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-1 votes
0 answers
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Inverse image of and $L^p$ function has null measure

Let $f \in L^{p}(\mathbb{R}^n)$ (maybe just mensarable). Is it true that $f^{-1}(\{c\})$, has zero Lebesgue measure, where $c$ is a constante. I know in the case $f$ is smooth this is true, because $...
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