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

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0
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1answer
12 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
3
votes
1answer
59 views

Total variation measure vs. total variation function

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on ...
1
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1answer
32 views

Measure theory : Lebesgue outer measure.

Let $E$ be a subset of $\mathbb{R}$ and $m^{*}(E)=0$. Prove that $m^{*}(E^{2})=0$, where $E^{2} = \{ x^{2} : x \in Ε \}$. Can you give me some ideas or hints? Thank you!
2
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1answer
25 views

If a measure $\mu$ is less than a measure $\nu$ on a generating $\pi$-system, can we conclude that $\mu \leq \nu$?

Let $\mu$, $\nu$ be finite measures on the non-degenerate compact interval $[a, b] \subseteq \mathbb{R}$ provided with the Borel $\sigma$-algebra. It is well-known that if $\mu(B) = \nu(B)$ for every ...
0
votes
1answer
24 views

Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
4
votes
1answer
84 views

$m(E)=0$ then $m(\lbrace x^2 : x\in E\rbrace$?

Let E be a subset of $\mathbb{R}$ with lebesgue measure zero. How can I prove that $\lbrace x^2 : x\in E\rbrace$ also has lebesgue measure zero? Let $\epsilon>0$, I should find a cover of ...
3
votes
1answer
31 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
1
vote
1answer
116 views

Change of variable formula for the image of a hypercube

Let $\varphi: \mathbb{R}^n\to \mathbb{R}^n$ be an injective $C^1$ map. Let $I=[0, 1]^n$. I want to show that $$m(\varphi(I))=\int_I \left|\det D\varphi(x)\right|dx.$$ This is a special case of the ...
0
votes
0answers
11 views

Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
3
votes
0answers
32 views

Theorem of Portmanteau: It suffices to show it for a base?

I have a question to the Theorem of Portmenteau, see here. Two equivalent statements to $P_n\to P$ weakly, are (1) $\limsup_n P_n(C)\leq P(C)$ for all closed sets $C$. (2) $\liminf_n P_n(O)\geq ...
9
votes
1answer
90 views
+200

Radon-Nikodým (write the density as a limit)

Let $\mu$ be a probability measure and $\nu$ a $\sigma$-finite measure on $(\mathbb{R},\mathcal{B})$ with $\nu\ll\mu$. Show that it is $\mu$-a.s. $$ \lim_{h\to 0}\frac{\nu [x-h,x+h)}{\mu ...
0
votes
0answers
9 views

Convolution of measures on a measurable group is associative

I've come across a statement in Kallenberg's Foundations of Modern Probability which claims this and only tells me to use Fubini's theorem. I am not very familiar with this topic and the text doesn't ...
0
votes
1answer
18 views

Prove that a complete field defines a partition of a set

Let $\Omega$ be arbitrary set. Let $Q$ be a partition of $\Omega$. I already proved that the collection of all unions of the cells in $Q$ is a complete field $\mathcal{F}$ (complete field is ...
4
votes
4answers
2k views

Translation invariant measures on $\mathbb R$.

What are all the translation invariant measures on $\mathbb{R}$? Except Lebesgue measure on $\mathbb R$ I didn't find any translation invariant measure. So I put this question? I know that if $\mu$ ...
-1
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1answer
31 views
-1
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0answers
19 views

A set which is not a G_delta nor an F_sigma [duplicate]

At first I thought of the Irrationals as a set satisfying these conditions but it does not.Is there any set with the above-mentioned properties?
0
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1answer
42 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuous stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
1
vote
1answer
34 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
1
vote
2answers
39 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
1
vote
2answers
168 views

Precise definition of random variables and probability measures

Suppose we have the probability space $(\Omega,\mathcal{A},P)$. Which of the following are right? $P$ is the probability measure defined on the events $\mathcal{A}$ as follows: ...
0
votes
0answers
30 views

How to separate self-defining values from sigma?

$$\sum_{k=1}^{m} \sum_{j=1}^{n} a_kx_j^{b_k+b_i} = \sum_{j=1}^{n} y_jx_j^{b_i}$$ What I need to do is solve $a$ for every $i$ given ($i$ is between 1 and $m$), so their result won't be composed of ...
0
votes
0answers
27 views

Does $L^p(L^1([0,1]))$ make sense?

I'm examining several function spaces like $L^p(X,\mu)$ where $X$ is a Banach space. Is it possible to take $X = L^1([0,1])$ and then look at $L^p(X)$? The problem I have is that I don't know ...
3
votes
1answer
46 views

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: ...
1
vote
2answers
34 views

Does the signed measure based on a Jordan decomposition of a function with bounded variation depend on the decomposition?

Let $g_1, g_2, h_1, h_2 : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous. Define $$ \begin{align} f_1 & := g_1 - h_1 \\ f_2 & := g_2 - h_2 \end{align} $$ and suppose ...
3
votes
2answers
90 views

Is the limit of measurable step functions measurable?

Let $(X, \mathcal M)$ be a measurable space where $\mathcal M$ is a $\sigma$-algebra on a set $X$. Let $f$ be a map $X \rightarrow E$ where $E$ is a metric space. Suppose $f$ is the pointwise limit of ...
-1
votes
2answers
61 views

Is a limit of measure of a sequence of sets equal to measure of limit of the sequence of sets?

I'm sitting at the same question desk as this: Limit of the measure of the converging sequence of sets. Actually, I can't prove it neither. PA6OTA gave a hint to show there is subsequence $A_{n_k}$ ...
2
votes
0answers
20 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
5
votes
0answers
43 views

Sets of Divergence for Fourier Partial Integals

It is a consequence of Carleson's theorem together with a transference argument that (see Section 4.3.5 in L Grafakos, Classical Fourier Analysis for proof) that the Fourier partial integrals of a ...
1
vote
1answer
60 views

Question about the measure induced from another measure, problem 1.22 from Folland

I'm self-learning Measure Theory using Real Analysis book of Folland. Unfortunately, I got stuck in this problem and couldn't find any clue to solve this. Can someone help me, or give me some hint so ...
2
votes
2answers
45 views

The irrational rotation is ergodic. The proof should use the idea of density point.

Consider $f_{\alpha}:S^{1}\rightarrow S^{1}$ the rotation of unit circle of angle $2\pi\alpha$, and let $\mu$ the Lebesgue measure in $S^{1}$. Let $\alpha$ irrational, show that $\left(f,\mu\right)$ ...
1
vote
1answer
19 views

Measure on Product Set

Consider a finite sequence of $\sigma$-finite measure spaces $(\Omega_i, \mathcal{F}_i, \mu_i)$. Constructing the product measurable space $$ (\Omega_1 \times \cdots \times \Omega_n, \mathcal{F}_1 ...
1
vote
1answer
29 views

Lebesgue integral of a ratio of Lebesgue densities

I need a hint to solve the following problem: $P$ is a probability mass on $\mathcal B(\mathbb R)$ with a Lebesgue density $h$, $f$ is another Lebesgue density. I need to show that $\int ...
0
votes
0answers
14 views

Connection regularity in measure theory and approximation in premeasure

In the measure theory lecture, we defined a measure-theoretic content as follows: $ \mu: \mathscr{C} \rightarrow [0,\infty]$ with the property being additive on disjoint sets and that the empty set ...
2
votes
1answer
34 views

What is a linear functional on continuous functions on the real line not given by a measure?

What is a positive linear functional on continuous functions on the real line not given by integration against a measure? I know that the dual of $C_c(\mathbb R)$ is the set of Radon measures, ...
1
vote
2answers
63 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
1
vote
2answers
103 views

Steinhaus Theorem in $\mathbb{R}^d$

I'm having some trouble proving the Steinhaus theorem in $\mathbb{R}^d$: Claim: Let $E\subset\mathbb{R}^d$ be a (measurable) set with positive measure. For some $\epsilon\gt0$, $E-E=\{x-y:x,y\in ...
16
votes
2answers
2k views

Steinhaus theorem (sums version)

This is a question from Stromberg related to Steinhaus' Theorem: If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval. I can't quite see how to modify the ...
0
votes
0answers
43 views

Dimension of a measure in terms of linear form on continuous functions

So this might be a bit of a weird question, but here goes. It is well-known (Riesz representation theorem) that the dual space of continuous functions on a compact $K$ identifies with the space of ...
0
votes
1answer
22 views

basic notions of measure theory: differences?

Could you help me differentiating the following notions of measure theory: law, probability, probability density, probability measure, probability distribution, distribution, distribution function. ...
7
votes
1answer
71 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
2
votes
1answer
43 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
3
votes
1answer
33 views

Joint Distribution Implies Independence…?

Consider the measurable space $(\mathbb{R}, \mathcal{B})$ and a probability space $(\Omega, \mathcal{F}, P)$. Define a finite sequence of random variables $X_1,\ldots,X_n: \Omega \to \mathbb{R}$. ...
0
votes
1answer
13 views

Equivalence of definitions of Gaussian Measure

Wikipedia's article on Gaussian measures notes this as the definition of Gaussian measures: $\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 ...
0
votes
0answers
12 views

Sunrise Lemma, left and right maximal functions

Would anyone be able to provide a proof of the following lemma: Where $| \cdot |$ is the Lebesque measure and $M_L$ and $M_R$ are the left and right maximal functions defined on $\mathbb{R}$ ...
-2
votes
1answer
66 views

Probability of a nonnegative submartingale converging to zero

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Some background information: ...
-1
votes
1answer
28 views

$E(f(|X_n|))$ property implies uniform integrability? [on hold]

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
2
votes
1answer
22 views

How does the Lebesgue measure measure non-cartesian product sets?

Consider the measure space $(\mathbb{R} \times \mathbb{R}, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda)$ where $\lambda$ is the Lebesgue measure on $(\mathbb{R}, \mathcal{B})$. Since ...
0
votes
0answers
35 views

Product Topology and Borel-$\sigma$-algebra

Let $S=\left\{1,2,...,n\right\}$ be equipped with discrete topology and let $X=S^{\mathbb{Z}}$. Then the so-called cylinder sets $$ [s_0,s_1,...,s_m]_n:=\left\{x\in X: \forall 0\leqslant i\leqslant ...
1
vote
1answer
22 views

Normally Distributed = Absolute Continuity?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. A random variable $X: \Omega \to \mathbb{R}$ is said to have the standard normal distribution if it has the density $f:\mathbb{R} \to ...
0
votes
1answer
83 views

proving that $\lim_{n\to \infty}P(A_n)$ exists and $\lim_{n\to \infty}P(A_n) =P(\lim\sup A_n)$ [on hold]

Borel-Cantelli lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ and $$\sum_{n=1}^\infty P(A_n) =\infty \Rightarrow P(\lim\sup A_n)=1$$ To show: If ...