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

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4
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1answer
18 views

A question about projections of product measure space

I am considering the space $\mathbb{R}^{\mathbb{N}}$ of real-valued sequences with the sigma-algebra $\mathcal{F}$ generated by sets of the form $$\{\omega \in \mathbb{R}^{\mathbb{N}} : \omega_k \in ...
1
vote
0answers
39 views

Intuition behind measurable random variables and $\sigma$-algebra

I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture. I'll try to ...
4
votes
2answers
118 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where ...
0
votes
2answers
46 views

The measure of set

Is it exist a closed set $E$ that is subset of $[a,b]$, $E \neq [a,b]$, which measure is $b-a$? I think, it exists, but can't find an example.
2
votes
0answers
29 views

Is $\phi$ convex if it is continuous and $\phi \Big(\frac{x+y}{2}\Big)\le \frac{1}{2}\phi (x) +\frac{1}{2} \phi (y)$ [duplicate]

Assume that $\phi$ is continuous real function on $(a,b)$ such that $$\phi\Big(\frac{x+y}{2}\Big)\le \frac{1}{2}\phi (x) +\frac{1}{2} \phi (y)$$ for all $x,y \in (a,b)$. Prove that $\phi$ is ...
0
votes
0answers
25 views

On the set of orderings generated by a measure

Let Ω be a finite set and let P denote the set of probability measures on the subsets of Ω. Fix a collection S of subsets of Ω and let L(S) denote the set of all linear orderings on S. Call a set ...
2
votes
1answer
40 views

Which one is larger, $\int^1_0 f(x) \ln f(x) dx\,$ or $\int^1_0 f(s) ds\, \int^1_0 \ln f(t) dt\,$?

If $f$ is a positive measurable function on $[0,1]$, which one is larger, $$ \int^1_0 f(x) \ln f(x) dx\,$$ or $$ \int^1_0 f(s) ds\, \int^1_0 \ln f(t) dt\,$$ I tried putting in some functions and ...
2
votes
1answer
33 views

Proving a Certain Planar Measure Is Zero on Horizontal Lines

Question: Suppose $\mu$ is a measure on $\mathbb{R}^{2}$ with respect to which all open squares are measurable. Suppose $\mu$ has the following property: there exists a constant $\alpha\geq 1$ ...
1
vote
1answer
39 views

$X\in L^1$, then $\int_{|X|>n}XdP\to 0$ and $P(A_n)\to 0 \Rightarrow \int_{A_n}XdP\to 0$

I'm trying to prove the following: 1. Suppose $X\in L^1$, then $\int_{|X|>n}XdP\to 0$ Attempt: $$\int_{\Omega}|X|dP = \int_{|X|≤n}|X|dP+ \int_{|X|>n}|X|dP = M<\infty \space \forall n$$ ...
1
vote
1answer
26 views

Measure-theoretic analogue of a result from elementary calculus

I recall from elementary calculus being taught about defining the "average" of a continuous function from a compact subset $K = [a, b]$ of $\mathbb{R}$ to $\mathbb{R}$ by $\frac{\int_{a}^{b} f(x) ...
0
votes
0answers
10 views

Image and preimage of sigma algebra and measurable functions

while learning the measure theory one reads a statement like this: Let f be a function from A to B and the corresponding sigma algebra of A. Then the image of the sigma algebra on A is the largest ...
2
votes
1answer
26 views

Proving Events belong to a tail sigma field

I'm really confused about tail sigma fields and how to prove that a set is or is not a tail event (belongs to the tail sigma field). I was wondering if anyone has seen examples of proving that a set ...
1
vote
0answers
23 views

Why $\int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn$?

I am reading the lecture notes. On page 5, formula (1.24) is $$ \int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn, $$ where $dn$ is the Haar measure on $N$, $U \subset N$ is ...
2
votes
1answer
71 views

Prove this liminf is a tail events

Let $A_{k}$,$k\geq1$ be [0,$\infty$)-valued random variables on a common probability space. I want to prove the following events are in/not in tail $\sigma$-field T($A_{k}$:$k\geq1$). First, event ...
0
votes
1answer
27 views

Sigma Finite Measure restricted to a small sigma-algebra is still sigma finite?

Let $(X,M,\upsilon)$ be a $\sigma$-finite measure, $N$ a sub-$\sigma$-algebra of $M$, then $\upsilon|_N$ is $\sigma$-finite measure in $(X,N)$?
1
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0answers
10 views

Exchangeable filtration $\mathcal{E}_n$ is the same as $\sigma(S_n,X_{n+1},\dots)$?

It should be a really intuitive conclusion but I kind of missing in the detail treatment. Suppose we have $\mathbb{R}^{\mathbb{N}}$ and equipped it with the sigma-algebra $\mathcal{B}_{\infty}$ ...
0
votes
1answer
32 views

Proving function is measurable

Define $f : [0, 1] → \Bbb R$ by $f(x) = 0$ if $x$ is rational,$1/(d^{1/2})$ if $x$ is irrational and $x = 0.0 . . . 0d . . . $, where $d$ is the first nonzero digit in the decimal expansion of $x$. ...
2
votes
1answer
17 views

Completion with respect a collection of measures.

Given a set of measures $\mathcal{M}$ on a measurable space $(\Omega,\mathscr{F})$, the $\mathcal{M}$-completion of $\mathscr{F}$ is defined as $$ ...
2
votes
1answer
80 views

Can Fatou's lemma and monotone convergence theorem be considered as equivalent?

Can Fatou's lemma and monotone convergence theorem be considered as equivalent?
0
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0answers
13 views

$\Sigma$-finite measure space which is a disjoint union of measurable sets of positive measure

Let $X$ be a measure space, $\mu$ its measure function. We assume that $X$ is $\sigma$-finite, i.e. it is a countable union of measurable subsets of finite measure. Suppose $X$ is a disjoint union of ...
2
votes
1answer
43 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
vote
1answer
113 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
45 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 ...
1
vote
2answers
43 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
votes
2answers
43 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
23 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
33 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
9 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 $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ ...
2
votes
1answer
33 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
vote
1answer
33 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
votes
1answer
27 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
vote
2answers
35 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
25 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
28 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
22 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
15 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 ...
0
votes
0answers
20 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
1answer
115 views

Rudin's Riesz Representation Theorem

The proof is from Rudin's Real and Complex Analysis. The Theorem states: Every open set $V$ satisfies $$\mu(E)=\sup\,\{\mu(K):K\subset E,\,K\,\,\text{compact}\}\,\,\,\,\,\,(3)$$ (Note: Here $\mu$ is ...
0
votes
2answers
28 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
vote
0answers
13 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
48 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
31 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
votes
0answers
36 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
24 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
19 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 ...
0
votes
0answers
24 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
52 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 ...