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Questions tagged [measure-theory]

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

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15 views

Is the image of a metric automorphism under a measure-preserving mapping also a metric automorphism?

I'm having my first tiny little bits of ergodic theory so please forgive the probable naivete of the question. So, I'm looking at the first page of chapter 8 of Cornfeld, Fomin, Sinai's "Ergodic ...
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9 views

Sigma-algebra generated by integer multiples

I came across this question and I'd like to check if my answer is correct. The problem is the following: Consider the collection $\mathcal{A}$ of subsets $A_1,A_2,...$ of $\mathbb{Z}$ such that $$A_i ...
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1answer
13 views

Does the following set have Lebesgue measure zero?

Let $F:\mathbb R^n\to\mathbb R$ is a non-constant continuous function. Is it true that $Leb[(x_1,...,x_n)\in\mathbb R^n:F(x_1,...,x_n)=0]=0?$ Here $Leb$ denotes Lebesgue measure. I don't know if this ...
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If $F(x)$ is continuously differentiable and its derivatives are bounded, then $F(X)$ is absolutely continuous

I know a result from measure theory saying that If $F(x)$ is differentiable at almost all $x\in [a,b]$, its derivative $f=F'$ is in $L^1[a,b]$, and $F(b)-F(a) = \int_{[a,b]}f(x)\,dx$, then $F$ is ...
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14 views

$ \mu $ premeasure and continousness - proof verification

Let be $ \mu $ a content on a Ring $R$ I want to prove following $$ (a) \leftrightarrow (b) \rightarrow (c) \leftrightarrow (d) $$ with a) $ \mu $ is a premeasure b) $\mu $ is continous "from ...
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2answers
28 views

Why do we need axiom of choice to prove that there does not a exist definition of $P(A)$, defined for all subsets $A \subset [0, 1]$

I'm reading "A First Look At Rigorous Probability" by Jeffrey S. Rosenthal. On chapter one there is a proof which I can't fully understand. Suppose, to the contrary, that $P(A)$ could be so defined ...
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41 views

Proof verification- $ \sigma $ - Algebra, Algebra, Ring

I want to prove following for a set $ X \neq \emptyset $ : $ M \subset P(X) $, where $ P(X) $ is the power set. 1) Any Ring is an Algebra 2) Any Algebra is a Ring 3) Any Ring is a $ \sigma $ -...
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15 views

Convergence of Bounded Variation on Open Subsets

I was reading a proof in Evan's Measure Theory and he stated a lemma without proof that I cannot justify: Let $U$ be an open connected subset of $\mathbb{R}^n$ Let $U_k := \{x \in U| \text{dist}(x, \...
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21 views

Is it true that $\mathcal H^{n-1}(\partial (A \cap B))=\mathcal H^{n-1}((\partial A) \cap B) + \mathcal H^{n-1}( A\cap ( \partial B))$?

Let $A$ and $B$ subsets of $\mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\mathcal H^{n-1} (\...
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1answer
19 views

Applying the dominated convergence theorem back and forth

Suppose $(X,\mathcal{M},\mu)$ is a measure space, $(f_{n})$ and $(g_{n})$ two sequences of integrable functions that tend to $f(x)$ and $g(x)$, respectively. Suppose also that $$|f_{n}g_{n}|\leq h_{1}...
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13 views

Haar measures are decomposable

The definition of decomposable measures is as follows: (Here $\mathcal{M}$ is a $\sigma$-algebra over $X$.) My question is part c) of the following exercise: I have managed to prove a) and b). For c)...
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1answer
45 views

Volume of image of closed rectangle in $\mathbb{R}^n$ under linear transformation

I am trying to solve the following problem: I believe I have done part (a). For $g_1$, and $g_3$(the first and third transformation shown in the image), it is easy to see that the image, $g_i(U)$ ...
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1answer
57 views

(Hint Needed) Real-Analysis Exam

So I'm trying to study for a qualifying exam and need a hint on the following problem. If someone knows of some useful lemmas and techniques, I would like to hear it! Let $(X,\mathcal{A},\mu)$ be a $\...
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1answer
23 views

Why is the set $\{x \mid f(x) \not= g(x)\}$ measurable in a topological hausdorff space?

assuming i have a measure space $(X, \mathscr{M}_X, \mu)$ and a measurable space $(Y,\mathscr{M}_Y)$, two measurable mappings $f,g:X \to Y$ and it holds that $Y$ is a topological hausdorff space with ...
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1answer
32 views

Measure Theory: Simple Measure Invariance Proof

Let M be a metric space, $f: M \rightarrow M$ be a measurable transformation and $\mu$ be a measure on M. Show that $f$ preserves $\mu$ if and only if $\int \phi d\mu =\int \phi \circ f d\mu$ for ...
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37 views

Understanding a function space

I was reading a paper on Homogenization theory, where the author uses the spaces of vector valued functions. Let us consider such a space $D[\Omega; C^\infty(R^n)]$, consisting of all the compactly ...
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10 views

Reference for “continuity in mean” of integrable functions

In the measury theory class I'm taking we proved the following theorem Let $f \in L^1(\mathbb{R^n})$. Then for almost every $x \in \mathbb{R^n}$ it holds that $$ \lim_{r\downarrow 0} \frac{1}{V_r}\...
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42 views

Is $\;F =\{∅, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$ a $\;\sigma$-algebra of $\;\{1, 2, 3\}?$

I think that it is not a $\sigma$-algebra of $\{1, 2, 3\},$ because for example $\{1, 2\} \cap \{1, 3\} = \{1\}$ $\{1, 3\} \cap \{2, 3\} = \{3\}$ $\{1, 2\}' = \{3\},$ etc. which are not elements ...
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1answer
22 views

How do I show that $\varphi(t)=(t,t^2,t^3)$ is a global chart

Let $f_{1},f_{2}:\mathbb R^{3} \to \mathbb R$, defined as $f_{1}(x,y,z)=x^2+xy-y-z$ $f_{2}(x,y,z)=2x^2+3xy-2y-3z$ and $M:=\{(x,y,z) \in \mathbb R^{3}:f_{1}(x,y,z)=f_{2}(x,y,z)=0\}$ I have shown ...
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Prove that $M:=\{(x,y,z) \in \mathbb R^{3}: z=xy\}$ is a $2-$dim $C^{1}-$Manifold

I am new to manifolds, and would like to understand whether I have the correct proof. As stated above, I want to prove that $M:=\{(x,y,z) \in \mathbb R^{3}: z=xy\}$ is a $2-$dim $C^{1}-$Manifold My ...
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11 views

Symmetrization of Conditional Expectation without common density

I know that the following rule for (factorizations of) conditional expectations is true: Theorem Let $X : \Omega \to \mathcal{X}=\mathbb{R}$ be an integrable, real valued random variable and let $Y:...
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41 views

Formal construction of the Cantor set $K$ and determination of $[0,1]\setminus K.$

Let $J_{0,1}:=[0,1]$. Step 1. We remove the central open interval $I_{0,1}=\big(\frac{1}{3},\frac{2}{3}\big)$. We denote with $J_{1,1}:=\big[0,\frac{1}{3}\big]$ and with $J_{1,2}:=\big[\frac{2}{3},...
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0answers
18 views

Egorov’s theorem: Small compact set

Let $K \subset \mathbb R^n, n \in \mathbb N$, be compact and suppose the measurable $f_n: K \to \mathbb R$ converge almost everywhere to a function $f: K \to \mathbb R$. By Egorov‘s theorem, for any $...
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1answer
29 views

How “big” can the boundary support of a continuous function be?

I was advised to place this part of my previous question (https://math.stackexchange.com/a/3083158/512018) into a new question, namely: Considering the factor $\partial \{x \in \mathbb R: f(x)\neq 0\...
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17 views

bernoulli distribution has a density with respect to dirac measure

We consider E={0,1} and the Bernoulli distribution $P_{\theta}$. We want to show that $P_{\theta}$ admit a density with respect to the measure µ= $\delta_0+ \delta_1$ I didn't understand the answer ...
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1answer
39 views

Have I understood Compact Set correctly

In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that $\overline{\{x \in \...
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0answers
17 views

Existence minimizer for total variation over measures

I want to prove that there exists a minimizer to the following problem $$ \min || \mu ||_{\text{TV}} \text{ such that } \mathcal{F} \mu = y $$ where $\mu \in \mathcal{M}([0,1])$, the space of Radon ...
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2answers
18 views

Is a continuous function of (integrable) Brownian motion always integrable?

Let $Z_{t}=e^{-W(t)}$ with $\{W(t):t\geq0\}$ a Brownian motion. Is $Z_{t}$ integrable, since it is a continuous function of Brownian motion? Furthermore, are all continuous functions of Brownian ...
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1answer
26 views

The vector space of Lebesgue measurable functions

I want to show that the set of all Lebesgue measurable functions is a vector space. Consider a measurable space $(X,\epsilon)$ and a measure $\mu$ on this space. Specifically, I wish to show that $\...
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1answer
27 views

Question regarding measurable set, Hausdorff-Space and almost everywhere properties of measurable functions f,g

I've been given the following task Let $(X,\mathscr{M}_X,\mu)$ a measure space. Two measurable mappings $f,g:X \to Y$ into a measurable space $(Y,\mathscr{M}_Y)$ are called equal almost ...
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1answer
34 views

How can I show $\int_{U}\frac{\partial f}{\partial x_{i}}(x)g(x)d\lambda^{d}(x)=-\int_{U}f(x)\frac{\partial g}{\partial x_{i}}(x)d\lambda^{d}(x)$

Let $U \subseteq \mathbb R^{d}$ and $f \in C_{c}^{1}(U)$ while $g \in C^{1}(U)$ Show that: $\int_{U}\frac{\partial f}{\partial x_{i}}(x)g(x)d\lambda^{d}(x)=-\int_{U}f(x)\frac{\partial g}{\partial ...
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1answer
21 views

Does this conclude that $ \int_E f_jg_j dx \to \int_E fgdx$?

Suppose $f_j \to f$ strongly in $L^2(E)$ and $g_j \to g \ $ weakly in $L^2(E)$, where $E \subset \mathbb{R}^d$ is measurable. Show that $ \ \int_Ef_jg_jdx \to \int_E fgdx$. Answer: I am quoting the ...
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0answers
19 views

Show that two Markov kernels almost surely agree

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E_i,\mathcal E_i)$ be a measurable space $X_1:\Omega\to E_1$ $X_2:\Omega\to E_2$ be $(\mathcal A,\mathcal E_2)$-measurable $\kappa$ ...
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1answer
19 views

A question about the outer measure generated

Let $\mathcal{K}\subseteq 2^{X}$ be a family of subset such that $\emptyset\in \mathcal{K}$ and let $\nu\colon\mathcal{K}\to [0,+\infty]$ an application such that $\nu(\emptyset)=0.$ We denote with $$\...
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1answer
52 views
+50

If $(κ_t)_{t≥0}$ is the transition semigroup of a continuous Markov process, is $t↦(κ_tf)(x)$ continuous for all bounded continuous $f$ and fixed $x$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $E$ be a metric space $(X_t)_{t\ge0}$ be an $E$-valued right-...
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1answer
54 views

What is the motivation of defining weak derivative as it is?

I've been reading lately about reproducing kernel Hilbert spaces (RKHS) and Gaussian processes (GP) and during my studies I came across with the concept of weak derivative and Sobolev spaces. I have ...
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31 views

Is there a non-zero measure on $\mathbb{R}^2$ such that all subsets of $\mathbb{R}^2$ are measurable?

1) Everywhere one can find many things about Lebesgue measure on $\mathbb{R}^2$. But what about arbitary measure? Is there a non-zero measure on $\mathbb{R}^2$ such that all subsets of $\mathbb{R}^2$ ...
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0answers
21 views

Definition of ergodic map

I ask the similar question before. About definition of Ergodic theorem. Now just sincerely ask another fundamental problem about the definition of ergodic map. The following definition is what I ...
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1answer
29 views

Question about Jordan Measure

I'm using the term "content" for Jordan measure and the term "measure" for Lebesgue measure. The definition of content that I was given is: A bounded set $D⊆ℝ^n$ has content if $x ↦ 1$ is Riemann ...
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1answer
27 views

Countable union of intersections of open subsets on $\mathbb{R}$ [on hold]

If we have a set $S=\{$intersection of any sequence of open subsets of $\mathbb{R}\}$. The textbook states $S$ is not closed under countable unions, but without any examples. Could anyone show me an ...
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2answers
24 views

(Prove or Disprove) If $X$ is a metric space, then the Borel $\sigma$-algebra ${\cal B}_X$ contains every countable subset of $X$.

I am tasked with determining whether the following statement is true or false: If $X$ is a metric space, then the Borel $\sigma$-algebra ${\cal B}_X$ contains every countable subset of $X$. I ...
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1answer
31 views

Probability of union using conditional probabilities

I'm struggling trying to calculate the probabilities associated with a random variable $Z$ whose value depends on the realizations of two other random variables $X,Y$. I have : $$ Pr(X=A)=0.25, Pr(Y=...
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1answer
53 views

Why is it legitimate to assume that the Chapman-Kolmogorov equations hold everywhere?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $E$ be a Polish space and $\mathcal E:=\mathcal B(E)$ $(X_t)_{t\ge0}$...
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0answers
69 views
+50

Rewrite $ \int_{\mathcal{S}}dP_X=1 $ as conditions on boxes in $\mathbb{R}^d$

Take $r\in \mathbb{N}$ and let $d\equiv r+\binom{r}{2}$. Consider a d-dimensional random vector $X\equiv (X_1,...,X_d)$. Let $P_X$ be the probability distribution of $X$. Assume that $$ \int_{\...
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0answers
14 views

Definition (and measurability) of sargmax

I am reading Statistical and computational trade-offs in estimation of sparse principal components by Wang et al. and in this paper, one can find the following definition (page 7): $$\hat v^k_{\max}(...
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2answers
17 views

Set of points where sequence of measurable functions is bounded from below and unbounded from above

We are given a space X with $\sigma$-algebra (of subsets of X) $\mathbb{F}$ and a sequence of measurable (w.r.t $\mathbb{F}$) functions $f_n: X \rightarrow \mathbb{R}$ for $n \in \mathbb{N}$. Now ...
2
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0answers
23 views

$\mathbb{Z}^2$-action on product is ergodic

I try to solve exercise 8.1.1 from Einsielder's book Ergodic Theory. It is: "Let $(X,B_X, μ, T)$ and $(Y,B_Y , ν, S)$ be ergodic Z-actions. Define a $\mathbb{Z}^2$-action on the product $(X×Y, μ×ν)$ ...
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1answer
19 views

Showing that the projection of a measure on path space onto marginals is continuous

Let $C([0,T], \mathbb{R}^d)$ be the metric space of continuous functions from $[0,T]$ to $\mathbb{R}^d$, endowed with the supremum metric. For any metric space $E$, let $\mathcal{P}(E)$ be the space ...
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1answer
19 views

Sigma Algebra Generated by Almost Sure Zero Random Variable

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X: \Omega \rightarrow \mathbb{R}$ be a random variable with $P(X = 0) =1.$ Let $\mathcal{N}$ be the class of sets $A \subset \Omega$ such ...
2
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1answer
59 views

Convergence of $\sigma-$algebra for converging stopping time

Given a filtration, ${\mathcal{F}_t},t\in[0,\infty).$ Let $T_n$ be a sequence of stopping time that converges to $T$ and $T_n\le T_{n+1}.$ We have correpsonding $\sigma-$algebra, ${\mathcal{F}_{T_n}}$ ...