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

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2
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1answer
12 views

Proofs of the Riesz–Markov–Kakutani representation theorem

Let $X$ be a compact Hausdorff space, $C(X)$ the set of all real continuous functions on $X$, and $\mathcal{B}$ be the Baire $\sigma$-algebra of $X$, which is the $\sigma$-algebra generated by the ...
1
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0answers
16 views

Sigma algebra generated by a quadratic function

I have a little difficulty determining what $\sigma(F)$ looks like for $$F: [0,1] \ni x \mapsto 1- |2x^2-1| \in [0,1]$$ I know that $F(x)=F(y) \iff x = \sqrt{1-y^2}, \ \ x,y \in [0,1]$. $\sigma(F) = ...
0
votes
2answers
33 views

Given a differentiable function, prove that the measure of the function on null set is $0$.

Suppose $f: [0,1] \to [0,1]$ is differentiable, that its derivative $f'$ is bounded, and that $A \subset [0,1]$ is a null set. Prove that $\mu(f(A)) = 0$. I was told to use Mean Value thm. But I'm ...
0
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2answers
33 views

Sigma algebra generated by an absolute value random variable

I need to find out what the sigma algebra generated by $Y$ looks like for $$Y: [0,1] \ni (\omega) \to 1- |2\omega -1| \in \mathbb{R}.$$ The graph of $Y$ is symmetric with respect to $\omega = ...
0
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2answers
40 views

Can the weight of something be a number with a repeating decimal?

This was brought up at my work today and I believe that it can be. My co-workers think that this is a crazy notion and I can't explain why weight can be a repeating decimal to their satisfaction. ...
1
vote
1answer
18 views

A kind of double expectation for conditional expectation

Is it true that for measurable $g \geqslant 0$ or $g \in L^1$, $$\int_{\Omega} \mathbf{E} [g (X, Y (\omega)) \mid Y = Y (\omega)] \mathbf{P} \left( \text{d} \omega \right) =\mathbf{E} [g (X, Y)]$$? ...
4
votes
0answers
31 views

subset of a compact set in $\mathbb{R}$ with nonempty interior has positive outer measure

Let $A\subset I=[a,b] \subset \mathbb{R}$, $a < b$ such that Int$(A) \neq \emptyset$. Show that $A$ has positive outer measure. What I have so far: Since Int$(A) \subseteq A$, by the ...
2
votes
0answers
8 views

Inward regular, not outward regular measure which is not locally finite

I want to find a measure, which is not locally finite, not outer regular but inner regular. Would the following example be correct? $\mu$ on the measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ ...
0
votes
0answers
12 views

a haar measure must have the property of invariance under reflection?

$G$ is a locally compact Hausdorff topological (multiplicative) group, $m$ is a (left) Haar measure on $X$. I have known that , for any $g\in{G}$, $m(gB)=m(B)$,but $m$ is invariant under reflection? ...
2
votes
1answer
31 views

Proving identities about measurable sets

You are given an interval $[a,b]$ (you can assume WLOG that $a<b$) and you take $A \subset [a,b]$ as a measurable set such that: $$\forall_{c,d\in Q} c\neq d \rightarrow (\{c\}+A) \cap (\{d\}+A))= ...
1
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1answer
27 views

Computing an outer measure

How do you actually compute an outer measure? I know the definition. It is: $$\mu^*(B):=\inf\left(\sum\limits_{k=1}^n \mu(I_k):B \subset \bigcup\limits_{k=1}^n I_k\right)$$ But how do you use this to ...
0
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0answers
19 views

Integration in complex measure

Let $v$ be a complex measure in $(X,M)$. Then $L^{1}(v)=L^{1}(|v|)$. I have made: $L^1(v)\subset L^1(|v|)$?. Let $g\in L^1(v)$ As $v<<|v|$ and $|v|$ is finite measure, then for chain rule, ...
1
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2answers
28 views

Hausdorff measure of a subset of $\mathbb R^3$

Let $f \in L^1_{\text{loc}}(\mathbb R^3)$. We define $A \subset \mathbb R^3$ as $$ A := \left\{ x \in \mathbb R^3 \, : \, \limsup_{r \to 0} \frac 1 r \int_{\mathbb B(x,r)} \vert f(y) \vert \, \mathrm ...
0
votes
2answers
22 views

Why is $\mu(\bigcap_{n=1}^{\infty}T^nX)=1$?

Let $\mu$ be an $T$-invariant measure on $(X,\mathcal{B})$. Then it is $$ \mu(T^nX)=\mu(T^{-n}T^nX)=\mu(X)=1. $$ Why is then $$ \mu(\bigcap_{n=1}^{\infty}T^n(X))=1? $$
2
votes
1answer
26 views

Product measure and measurability

Let $(X,\mathcal{G})$ and $(Y,\mathcal{H})$ be measure spaces, and $f:X\times Y\rightarrow \mathbb{R}$ be measurable with respect to the product measure space $(X\times ...
1
vote
1answer
21 views

Cadlag process and measurability.

Let $(\Omega,(\mathcal{F_t})_{t\geq0},P)$ be a filtered probability space and $X=(X_t)_{t\geq0}$ a real-valued adapted cadlag process. Let $A\subset\Omega$ (resp. $B\subset\Omega$) be the event that ...
1
vote
1answer
28 views

Proof of replacement rule in conditional probability

The answer to this question gives a replacement rule for conditional probability. But how do you prove this? I tried integrating both sides w.r.t. $P_X$ and fiddling around but it didn't get me ...
0
votes
0answers
14 views

Integrability of Banach-space-valued functions

In http://en.wikipedia.org/wiki/Dominated_convergence_theorem it says that: "The dominated convergence theorem applies also to measurable functions with values in a Banach space, with the dominating ...
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0answers
19 views

Borel sigma-algebra vs Toplogical space on R

i got two questions on measure theory: The Borel sigma algebra on R is not containing all the toplogical space on R. At the other hand it is generated by all the open sets of the topological space. ...
0
votes
2answers
24 views

Proof $x\to ||f(x)||_B \in \mathbb{L}^1(X,S,\mu,\mathbb{R})$

Why is the function $g:x\to ||f(x)||_B$ in $\mathbb{L}^1(X,S,\mu,\mathbb{R})$, where $f\in\mathbb{L}^1(X,S,\mu,\mathbb{R})$, and $||\cdot||_B$ is the norm in the Banach space that $f$ maps into? I ...
1
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0answers
12 views

Expectation with respect to empirical distribution

Let $(\Omega,\mathcal{A})$ be a measure space and $X$ a random variable with distribution $P$. The expectation of some measurable function $g$ with respect to $P$ is $$ \mathbb{E}_P[g(X)] = ...
2
votes
3answers
29 views

Family of functions that are bounded in $L^1$ but *NOT* Uniformly Integrable

I'm having a difficult time constructing a counter example to this. My intuition (sloppily) is to construct a family of functions {$X_n$} that have Dirac pulses at $n$ and $-n$. Such that $\sup_n \Bbb ...
2
votes
1answer
25 views

Hausdorff dimension of a Sierpinski-like triangle

Define the set $A \subset \mathbb R^2$ by proceeding as follows. Let $A_0$ be a closed equilateral triangular region of side 1. $A_1$ are the three equilateral triangular regions of side $\frac 1 3$ ...
0
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0answers
34 views

Does this still give the measurable sets of an outer measure?

If we define the collection of measurable sets of a measure to be the maximal collection of sets such that $\mu(A) = \mu(A\backslash E)+\mu(A\cap E)$, where $A,E$ are a sets in the collection$-$ is ...
1
vote
1answer
19 views

Disjoint union of uncountable measurable sets of positive measure

Let $X$ be a measure space, $\mu$ its measure function. Suppose $X$ is a disjoint union of a family of measurable sets $\{X_\alpha : \alpha\in A\}$. Suppose $\mu(X_\alpha)\gt 0$ for all $\alpha\in A$. ...
1
vote
0answers
16 views

Heuristic: Daniell integral vs. Lebesgue integral

What are the advantages of the Daniel Integral over the Lebesgue integral and visa-versa? Heuristically speaking, I was wondering why this axiomatic operator is less popular besides the fact that it ...
1
vote
1answer
9 views

Augmenting a filtration

I have a short question regarding the topic in the title. Let $(\mathcal{F}_t)$ be a filtration on some probability space. Let $(B_n)$ be a sequence of events such that$B_n \in ...
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0answers
16 views

Admissible transference plans, characterizing a condition on measures.

In Villani's, Topics in optimal transportation, pg.18 we find that for a borel measure $\pi\in X\times Y$, where $(X,\mu)$ and $(Y,\nu)$ are probability measure spaces, the condition \begin{equation} ...
0
votes
1answer
17 views

Integral and dominated convergence theorem

Let us define $g_n(x)= n\chi_{[0,n^{-3}]}(x)$. I am looking for help to answers the following problem $(a)$ Show that if $f$ $\epsilon$ $L^2([0,1])$ then $\int_0^1f(x)g_n(x)dx \rightarrow 0$ as $n ...
2
votes
2answers
51 views

Amazing property of martingales

let $Y_1,Y_2,..$ be a sequence of equally distributed, independent and positive random variables. Consider $X_n = Y_1…Y_n$. Under which condition is $X_n$ a (super)-martingale? Show that neglecting ...
1
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2answers
55 views

A continuous bijection from the Cantor Set to [0,1]

If $C$ is the Cantor Set, I am asked to show that there exists a continuous bijection, say $f$, that maps $C \to [0,1]$. My best guess thus far has been the Cantor Function, however (using this ...
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0answers
15 views

Regularity of product measure

Suppose $X$ and $Y$ are compact Hausdorff spaces and $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are finite regular Borel measure spaces. (By regular I mean that every measurable set can be ...
1
vote
1answer
17 views

Topology of weak convergence, linear functionals and probabilistic intuition

One very basic question regarding the topology of weak convergence. We know that given the following: $X$ metrizable topological space, $\mathcal{B} (X)$ Borel $\sigma$-algebra, $\Delta (X)$ ...
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0answers
18 views

My question is in measure theory in convergence theorems [duplicate]

My question is in measure theory in convergence theorems We know increasing convergence theorem and its proof , its proof is so easy and there is no problem but someone told me you can prove it by ...
0
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0answers
43 views

A problem from Folland's Real Analysis Book [on hold]

If $E $ is a Borel set in $R^n $,then $D_E (x) =1 $ for almost all $ x \in E $ and $D_E (x)=0 $ for almost all $ x \in E^c $ where $ D_E (x)= \lim_{ r \to 0} m(E \cap B(x,r))/m(B(x,r)) $ This Is my ...
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0answers
11 views

Interpretation of $\sigma$-algebra and filtrations (follow-up question)

This is a follow-up question to Interpretation of sigma algebra, particularly to Jun Deng's excellent answer. He used the example of two coin tosses to explain some fundamentals of how filtrations and ...
7
votes
2answers
177 views

Do two closed subsets of $[0, 1]$ with measure $\frac{1}{2}$ intersect?

Let $A$ and $B$ be two closed subsets of $[0,1]$, each with a length of $1/2$. Is it always true that $A\cap B\neq \emptyset$? My intuition is yes, because: Either they intersect in their interior; ...
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votes
1answer
34 views

Problems on Integration By Parts for Lebesgue Measures

If $F $,$G $ are absolutely continuous function on $[a,b]$, prove that so is $FG $ and $$ \int_{a}^{b} (FG' + GF')(x)dx$$ $= F(b)G(b)-F(a)G(a) $ This is my homework problem.I can't solve this ...
0
votes
1answer
43 views

Haar measure on locally compact group

Please I need a help to solve two problems in the book of principles of Harmonic analysis of Deitmar and Echterhoff Exercise 1.4 Let $G$ be a locally compact group with Haar measure $\mu$, and let ...
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0answers
17 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
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votes
1answer
27 views

One question about measure theory [on hold]

Let $(X,S,μ)$ be a finite measure space $μ(X) < \infty $ and $α$ is a finite positive measure on $S$ If $α(A)=\int_{A}{}hdμ$ where $ h \in L^1(μ)$ Prove that $α<<μ$
4
votes
0answers
24 views

Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.

Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure. My work: I proved the ...
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votes
1answer
14 views

Visualizing a probability measures through a probability density functions

I found a previous question with a very nice answer, but still there is something that is not completely clear to me. We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let ...
0
votes
1answer
20 views

Calculate the Lebesgue integral of a step function

I'm having some trouble with this problem. Let $$f(x)= \begin{cases} 1 &\text{for}\,\, x = \frac{1}{n}\,,\, n=1,2,\cdots \\ 2 &\text{otherwise} \end{cases}$$ Compute the value of the ...
2
votes
1answer
16 views

Independence of sigma algebras of sigma algebras

I have a bunch of questions all of which more or less fall under the subject in the title. The first one goes as follows. Let $E_1,E_2,\ldots,E_n$ be collections of measurable sets on ...
0
votes
1answer
18 views

Can an indicator function be a valid Radon Nikodym derivative?

Take a process $X_t$ defined on a canonical space with probability $\mathbb{P}$. Can the indicator function $\mathbb{1}_{X_t< U}$ be a Radon Nikodym derivative? That is can we have a measure ...
3
votes
1answer
74 views

Showing the almost everywhere equality of two unions of sets

Sorry for the ambiguous title. I am trying to prove this seemingly simple statement. Let $\{A_i\}$ and $\{B_i\}$ be sequences of measurable sets in a measure space $(X,\mathcal{A},\mu)$ such that ...
1
vote
1answer
44 views

almost sure convergence for non-measurable functions

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume for each $n$, $Y_n:\Omega\rightarrow\mathbb{R}$ is a function but $Y_n$ is not necessarily $\mathscr{F}$-measurable. In this case, is it ...
2
votes
1answer
34 views

Do the eigenvectors of a random orthogonal matrix have Haar measure?

For orthogonal $Q$ with Haar measure, does the group of unitary matrices $U$ which diagonalize $Q=U\Lambda U^H$ have Haar measure? I'd be happy to know any answer, even if it's only true for certain ...
1
vote
1answer
25 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...