Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
20,989 questions
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0answers
9 views
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 ...
2
votes
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 ...
0
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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 ...
0
votes
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)=\...
0
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3answers
31 views
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 ...
1
vote
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-...
1
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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 $(X, \mathcal{M}, \mu)$ ...
3
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0answers
18 views
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 $(\...
1
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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 \...
2
votes
0answers
37 views
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 $+\...
3
votes
2answers
60 views
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 ...
-1
votes
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 ...
1
vote
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}^\...
2
votes
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 ...
0
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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:
...
0
votes
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 ...
0
votes
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)=\...
1
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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 ...
1
vote
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,
$...
1
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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 ...
1
vote
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: \...
2
votes
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 ...
1
vote
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\...
1
vote
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$. ...
0
votes
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 ...
0
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0answers
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 \...
0
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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 ...
2
votes
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$ $\...
2
votes
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 ...
1
vote
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 \...
0
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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 ...
0
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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\}$ ...
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 ...
3
votes
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 ...
-1
votes
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=\...
0
votes
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)&...
1
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0answers
31 views
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 ...
0
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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 ...
1
vote
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)$ ...
0
votes
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
11 views
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.
2
votes
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 ...
0
votes
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 ...
0
votes
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(...
4
votes
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 $\mathbb{R}^d$ ...
1
vote
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 ...
1
vote
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
votes
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([...
1
vote
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 ...
