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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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Finding the generator of a $\sigma$-algebra

I was looking at this post, and had a couple questions regarding it. 1.) Given $\Omega$ ={1,2,3,4}, $\mathcal F = 2^{\Omega}$, how can I find a $C$ such that $\sigma (C) = \mathcal F$? I know it is ...
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1answer
16 views

Condition of being a measure

I'm working on a measure theory problem and I'm completely stumped. I'm trying to find out for which integers $j$, $\mu$ will be a measure on $(\mathbb Z_+, \mathcal P(\mathbb Z_+))$ where $\mu$ is ...
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0answers
7 views

Singular function that is Holder for all $\alpha<1$

I am asking for an example of a singular continuous function that is Holder for all $\alpha<1$. We know that such function cannot be Lipschitz, otherwise it is absolutely continuous. We also know ...
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1answer
24 views

If $\omega\in \limsup A_n$, then $\sum_{n\geq 0}\Bbb{1}_{A_n}(\omega)=+\infty$

I want to show that $\omega\in \limsup A_n\implies\sum_{n\geq 0}\Bbb{1}_{A_n}(\omega)=+\infty$ where $\{A_n\}_{n\geq 0}$ is a sequence of subsets of $\Omega$ and \begin{align} \Bbb{1}_{A_n}(\omega)=\...
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3answers
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Does there exist a Lebesgue measurable subset $A$ of $R$ such that for every $a<b$ we have $m(A\cap(a,b))=(b-a)/2$?

Does there exist a Lebesgue measurable subset $A$ of $R$ such that for every $a<b$ we have $m(A\cap(a,b))=(b-a)/2$? I searched before posting and found a similar question here, but it isn't ...
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1answer
37 views

Understanding Caratheodory's theorem

Let $\mu^{\star}$ be an outer measure on $X$. The book "Real Analysis" by Folland motivates the definition of set $A$ being $\mu^\star$ measurable as follows. It says first that If $E$ is a "well-...
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2answers
33 views

Measure theory homework question with a set of infinite measure and subsets of finite measure.

I've searched and haven't seen this problem in my searches. Admittedly I may have missed it and I apologize if that's the case. The problem statement is the following: Let $(X, \mathcal{M}, \mu)$ ...
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Measurability of random variable wrt pullback $σ$-algebra

Three Polish spaces $A$, $B$ and $R$ are given, and are equipped with their Borel $σ$-algebras $\mathcal B_A$, $\mathcal B_B$ and $\mathcal B_R$. Let $f\colon A \to B$ and $g\colon A \to R$ be $(\...
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1answer
13 views

Union of infinitely often events

I have a simple question. Is the following reasoning correct? $A_n \subseteq B_n \cup C_n \;\forall\;n \Rightarrow [A_n i.o.] \subseteq [B_n \cup C_n i.o.]$ $\Rightarrow \mathbb{P}([A_n i.o.]) \leq \...
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0answers
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Group action and invariant measures

Let's first start with the example from where this question arose. I consider the metric space $\mathbb{R}$ and the Lebesgue-measure $\lambda$ on $\mathbb{R}$ as well as the transitive group action $+\...
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2answers
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Does there exist a measurable subset $A \subset \mathbb{R}$, such that $\mu(A)$ is finite, but $\mu(\{a+b|a,b\in A\}) = \infty$?

Does there exist a measurable subset $A \subset \mathbb{R}$, such that $\mu(A)$ is finite, but $\mu(\{a+b|a,b\in A\}) = \infty$? Here $\mu$ stands for Lebesgue measure. If such subset exists, it can ...
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1answer
37 views

A bounded set is Lebesgue measurable iff its Lebesgue outer measure equals Jordan inner measure

Added: It was pointed out that this proposition is false. Let $A$ be a bounded set in $\mathbb{R}^n$. How to prove that $A$ is Lebesgue measurable iff its Lebesgue outer measure $m^*(A)$ equals its ...
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1answer
17 views

Conditions for measure being finite and $\sigma$-finite

working on a problem for measure theory and wanted to see if I'm on the right track. I've shown that over a measurable space $(X,\mathcal M)$, $\sum_{j=1}^\infty a_j\mu_j$ where {$\mu_j$}$_{j=1}^\...
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1answer
14 views

Counterexample for the $L^p$ Ergodic Theorem of Von Neumann when $p=\infty$

I am looking for a counterexample to the following theorem when $p=\infty$: $L^p$ Ergodic Theorem of Von Neumann. Let $1\leq p<\infty$ and let $T$ be a measure-preserving transformation of ...
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2answers
20 views

I want to get ratio of x to y in this equation: $x + 2*(x/10)^2 + 3*(x/10)^3/10 + 4*(x/10)^4/100 + … + n*(x/10)^n/10^{n-2} = y$

I need the ratio of x to y in this equation: $\displaystyle \sum ^{\infty }_{n=1} n\frac{\left(\frac{x}{10}\right)^{n}}{10^{n-2}} =y$ or: ...
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0answers
42 views

Outer and inner measures

I want to prove that $m_{∗}(A \cap (a, b)) = (b − a) − m^* (A^c \cap (a, b))$. Since $(a,b)$ is open, it is measurable. Then $(b-a)=m^*((a,b))=m_*((a,b)$. Also by the definition of measurable ...
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0answers
15 views

Measure obtained by pushing forward a constant function

working on a problem for measure theory, and I'm a bit stuck. Let $f:X \to Y$ be a mapping, and suppose $(X, \mathcal M,\mu)$ is a measure space. Given the push forward of $\mu$ by $f$: $$f_*\mu(E)=\...
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1answer
17 views

Preimage of generated $\sigma$-algebra and the $\sigma$-algebra generated by the preimage

I'm working on proving the equality between the preimage of generated $\sigma$-algebra and generated $\sigma$-algebra of preimage. I noticed that there's a duplicate: Proof that the preimage of ...
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2answers
30 views

uniform convergence of $f_n = n \chi_{[1/n, 2/n]}$

$f_n = n \chi_{[1/n, 2/n]}$. Show that $f_n$ converge uniformly to $0$. $f_n = n$ if $1/n \le x \le 2/n$, and $0$ otherwise. But, I am a little bit confused here. When $n$ goes to infinity, $...
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0answers
11 views

A Brief Clarification on Closed set Automatically Being Lebesgue measurable

This maybe a non-sense question -- but just to make sure: Suppose I prove something about some arbitrary set in $\mathbb{R}$ and find that the set is closed. Because I know the set is closed, it's ...
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2answers
20 views

Co-countable set and a countable set

I have this question: let $X$ be an uncountable set. Let $\mathcal{M}=\{A \subseteq X : A $ is countable or A is co-countable$\}$, co-countable means its complement is countable. Then, define $m: \...
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1answer
31 views

Associativity of the product of measurable spaces

The product of two measurable spaces $(X_1,M_1)$, $(X_2,M_2)$ is defined to be $(X,M)$, with $X = X_1\times X_2$, cartesian product of $X_1$ and $X_2$ and $M = M_1 \otimes M_2$ sigma-algebra generated ...
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1answer
26 views

Prove that $X$ is measurable if and only if the restrictions of $f$ to $A$ and $B$ are measurable.

I have that $X$ is a measure space, i.e. there is a sigma algebra $\mathcal M \subset P(X)$ and a measure $\mu$. I am trying to prove the following but I am having a hard time. Assume that $X=A\...
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0answers
18 views

Computing Measure of a Cantor Set's Image

I'm working on a problem involving a constructing a homeomorphism that takes the cantor set to a set of positive measure in the image. I'm having trouble showing the image has Lesbugue measure $1$. ...
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0answers
49 views

Why is the set of continous paths of a browian motion not measurable?

Øksendal states in his book "stochastic differential equations" (Defintion 2.2.1 iii)) that the set $H = \{\ \omega \mid t → B_t (\omega)\ \text{is continuous}\ \}$ is not measurable with respect ...
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21 views

$\pi-\lambda$ Theorem and Monotone Class Theorem

In Billingsley, he defines (and proves) the two related theorems: $~$ $\pi-\lambda~$ Theorem. If $\mathscr{P}$ a $\pi$-system and $\mathscr{L}$ a $\lambda$-system, then $\mathscr{P} \subset \...
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1answer
20 views

If ($f_n$) is Cauchy in measure, there exists a subsequence ($g_k$) of ($f_n$) such that ($g_k$) converges a.e.

This is the proof that if $(f_n)$ is cauchy in measure, there exists a subsequence $(g_k)$ of $(f_n)$ such that $(g_k)$ converges a.e. (from Bartle). I just don't understand one step in this proof. I ...
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1answer
35 views

Prove the existence of a member A of a sigma algebra whose outer measure is equal to the measure of a subset of A

Let $\mu$ be a measure on a semi-ring $A$ $\subseteq$ $P(\Omega)$ and let $\mu^*$ be the outer measure generated by $(A, \mu)$. Prove that for any $E$ $\subseteq$ $\Omega$ $\exists$ $B$ $\in$ $\...
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4answers
62 views

If a set is closed under countable unions, is it closed under countable intersections?

I am trying to think through the intuition of DeMorgan's laws. If I have a set $\Omega$ and a sequence of subsets ($A_n$)={$X_1$, $X_2$, ...} how can I know that given the fact $\Omega$ is closed ...
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1answer
34 views

Folland Chapter 2 Exercise 7

I'm working on the problem linked here: Measure Theory - Folland - Problem 2.7 For completeness, I restate it here: Suppose that for each $\alpha \in \mathbb{R}$ we are given a set $E_{\alpha} \in \...
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1answer
32 views

Excersise 5H Bartle

Suppose that $f_{1}$ and $f_{2}$ are in $L(X,\Omega, \mu)$ and let $\lambda_{1}$ and $\lambda_{2}$ be their indefinite integrals. Show that $\lambda_{1}(E)=\lambda_{2}(E)$ for all $E\in \Omega$ if and ...
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0answers
19 views

The Bayes optimal predictor is optimal

I'm attending a lecture in Machine Learning Theory, and since I have almost no background in Probability Theory, I'm having problems with the following exercise: Let $X$ be a set and $Y = \{0, 1\}$ ...
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0answers
27 views

Exercise 5K (Elements of integration-Bartle)

Let $X=\mathbb{N}$, let $\Omega$ be all subsets of $\mathbb{N}$, and let $\mu$ be the counting measure on $\Omega$. Show that $f$ belongs to $L(X,\Omega, \mu)$ if and only if the series $\sum f(n)$ is ...
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2answers
51 views

Infinite dimensional Banach lattice $L^\infty(X)$ is not order continuous

Consider an arbitrary measure space $(X,\Sigma,\mu)$, with the only assumption being that $L^\infty(X)$ is infinite dimensional. Consider $L^\infty(X)$ as a Banach lattice with the usual ordering. As ...
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1answer
35 views

Prove that a set of functions that converges is in sigma-algebra

I'm new at measure theory and convergence, so bare with me. I have some trouble with the following exercise: "Let $X_1,X_2,...$ be random variables on $(\Omega,\mathcal{F},P)$. Show that the set $A=\...
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1answer
9 views

For finitely additive functions: upper-semicontinuous $\Rightarrow$ pre-measure

I want to prove that a finitely additive function that is upper-semicontinuous is a pre-measure, when considered on a semi-ring. That is, when $A_i\in R,$ $\mu(A_1 \cup A_2\cup...\cup A_n)=\mu(A_1)+.....
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2answers
30 views

Can't follow Proof in Folland: Outer Measures and Pre-measures.

The following is given in Folland: I don't understand why $u_0(E) \leq u^*(E)$. I thought that since $u^*(E)$ is the inf of $\sum_{n=1}^{\infty} u_0(A_j): Aj \in A, E \in \bigcup A_j$ that $u^*(E)&...
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0answers
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Semi-Inclusion-Exclusion Theorem for an Outer Measure

Let $\mu$ be a measure on a semiring $A \subseteq P(\Omega)$ and let $\mu ^*$ be the outer measure generated by $(A, \mu)$ Prove the semi-inclusion-exclusion principle. $\mu ^*(E \cup F) + \mu ^*(E ...
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1answer
32 views

For which sets $X$ is the counting measure on $(X,P(X))$ $\sigma$-finite?

I'm trying to prove that the counting measure on $(X,P(X))$ is $\sigma$-finite if and only if $X$ is countable as an answer to the title. Intuitively, it feels that $X$ should be countable so I ...
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1answer
20 views

Monotone Class closed under complementation.

Are Monotone classes always closed under complementation? I attempted to construct a counterexample however I was not able to do so. Recall the definition of a monotone class is: $C \subset P(X)$ ...
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1answer
26 views

Proving algebras are closed under set difference

Using De Morgans rule I can easily prove $E\cap F$ is an element of $\mathcal{A}$. This is done by applying the following definition and the three axioms. Definition: Let $X$ be some set. An algebra ...
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0answers
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sigma field of infinite product space

I am interested to know if the random function {X(t):t>=0,X(t) ϵR} is a set belonging to the product sigma field (∏_(t≥0)▒R_t ,∏_(t≥0)▒B_t ), B_t is Borel field in R_t=R.
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1answer
66 views

Cantor set to show that the Borel measure is not complete

I would like to use the Cantor set to illustrate the fact that the Borel measure is not complete. To do so, I saw different sources (wiki and math.stackexchange, if I understood well) using the ...
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1answer
21 views

Combining two integrals with respect to different measures

Suppose $\mu_1$ and $\mu_2$ are finite and non-negative measures. I am curious if, given those two assumptions, is it always the case that $\int_{x \in A} f(x) d\mu_1 (x) + \int_{x \in A} f(x) d\mu_2 ...
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0answers
30 views

Expectation of function under bounded cdf

Suppose that $F(x)$ is a cumulative distribution function such that $F(x)\geq G(x)$ for all $x$ for some function $G(x)$. If furthermore for some bounded, continuous function $h$ $$ \int \int \int_A h(...
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4answers
96 views

The notion of limsup for sets

I was working through The Borel-Cantelli lemma from Real Analysis problem book and ran into the following comment: Let $\{E_k\}_{k\geq1}$ is a countable family of measurable subsets of $\mathbb{R}^d$ ...
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0answers
21 views

Probability Distribution on Power Set of Uncountable Set

I've been thinking for a while about the following question, and I cannot make up my mind. Any ideas are welcome about it: I have a set function $V:\Omega\to 2^{[0,1]}$, and ideally, I would want to ...
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0answers
36 views

Differences of indicator functions defined on sets that are “close” to each other

Let $x \in D$ where $D$ is a compact domain. Consider a smooth function $f(x)$ and another function $f_n(x)$ (a sequence of functions indexed by $n$) such that $f_n \rightarrow f$ uniformly as $n \...
3
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1answer
26 views

Lebesgue measure on countable union

Given the Borel set $A=\cup_{n=1}^{\infty}[n,n+1/n]$ how do I find its Lebesgue measure? My attempt: Given $\lambda$ we have that \begin{align*} \lambda(\cup_{n=1}^{\infty}[n,n+1/n])&=\lambda([...
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1answer
31 views

How to show a subset of $\mathbb{R}^2$ is a Borel set?

How to show the following subset is a Borel set? $$ \{(x,y)\in \mathbb{R}^2:y=x,0\leq x \leq 1\}. $$ In How to show this set is a Borel set?, the answer is to write the set as intersection of an open ...