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

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3
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1answer
25 views

an exercise about changing the measure and convergence in $L^1$

this is exercise 17.12 from probability essentials written by jacod & protter. Suppose $lim_{n→∞} X_n = X$ a.s. Let $Y = sup_n |X_n − X|$. Show $Y < ∞$ a.s. , and define a new probability ...
0
votes
0answers
8 views

Homogeneous Markov chains with general state space

I found in the book Markov Chains by Revuz the following definition of a Markov chain. In the following $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables on a probability space ...
0
votes
1answer
12 views

If $\lambda(A_n \cap A_k) = 0$ then $\lambda \left( \bigcup_{n=0}^{\infty} A_n \right) = \sum_{n=0}^{\infty} \lambda(A_n)$

Let $A_n$ be borel set such that $\lambda(A_n \cap A_k) = 0 \quad \mbox{for} \quad n\neq k$. $\lambda$ is Lebesgue measure. Show that $$\lambda \left( \bigcup_{n=0}^{\infty} A_n \right) = ...
3
votes
0answers
21 views

Weak convergence of measures iff subsequence of subsequence of distribution functions converges a.e.

I'm trying to prove the first part of Proposition 8.1.8 in V.I.Bogachev, Measure Theory 2: A sequence of signed measures $\mu_n$ on the interval $[a,b]$ converges weakly to a measure $\mu$ ...
1
vote
1answer
10 views

Squared Hellinger Distance subadditive for Product measures

How can I show that the squared Hellinger Distance is subadditive for Product measures? We have $\mathbb{P} = \otimes_{i=1}^n \mathbb{P_i}$ and $\mathbb{Q} = \otimes_{i=1}^n \mathbb{Q_i}$ ...
2
votes
1answer
15 views

A question on Characteristic function on Cantor like set

Consider a Cantor like set $C$ with measure $1>\epsilon>0$ on the interval $[0,1]$. Is it possible to find a measurable set $F \subset [0,1]$ with $m(F)=1$ such that $\displaystyle \chi$$_c|_F$ ...
0
votes
0answers
15 views

$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n)$

I am trying to prove the equality $$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n),$$where $\mathcal B(\mathbb R^i)$ is the Borel $\sigma$-algebra on $\mathbb ...
2
votes
0answers
13 views

Poisson Random Measure

I'm really new to this area of random measures, and I'm a bit confused on how to get started on this problem. Let $\mu$ be a measure with on $\mathbb{R}$ with $\mu(\left\{0\right\}) = 0$ and ...
1
vote
1answer
43 views

Measure concentrated at a point

What does "a finite random measure $\nu$ is concentrated at a point" mean? And in this case, what is equal to $\int_{\Omega} x d\nu$ ? Thank you.
2
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0answers
30 views

An exercise on conditional expectation and some related questions.

I tried to solve an exercise involving conditional expectations, and in doing so some question's popped up in my mind. First the exercise: $|Z| \le c \textrm{ P.-a.s.} \Rightarrow |E\{ Z | ...
2
votes
0answers
24 views

Convergence of a subsequence of a subsequence of distribution functions

I'm trying to find a solution for the following problem: Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of signed (Baire)-measures (of bounded variation) on $[a,b]$ and let $F_{\mu_n}(t):=\mu_n([a,t))$ ...
5
votes
2answers
66 views

Does $\mu(x+B)= \mu(B)$ for all balls $B$ imply that $\mu$ equals the Lebesgue measure (up to scaling)?

Suppose that $\mu$ is a measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ such that $\mu(K)<\infty$ for any compact set $K$ and $$\mu(x+B) = \mu(B) \tag{1}$$ for all $x \in ...
1
vote
1answer
43 views

A question about integrable function.

My question: Let $(\Omega, \mathscr{U}, \mu)$ be a measure space, and let $X$ be an integrable function and let $A, \ \ \{A_n\} \in \mathscr{U}; n\in \Bbb N$. How to prove that $$\int_{A_n} X d\mu ...
0
votes
2answers
38 views

Is $t\mapsto 1_{[0,t]}(s)$ for a fixed $s\ge 0$ continous?

Let $s\ge 0$ and $$f:[0,\infty)\to\left\{0,1\right\}\;,\;\;\;t\mapsto 1_{[0,t]}(s)$$ Is $f$ continuous at $t_0\ge 0$? If $s>t_0$, then $f(t_0)=0=\displaystyle\lim_{n\to\infty}f(t_n)$ for all ...
1
vote
0answers
24 views

A problem similar to $L^2$ Fourier transform, but in the setting of Borel measure.

Problem: Let $\mu$ be a finite Borel measure on the real axis, supported on a countable set $\mathbb{Q} \subset \mathbb{R}$ (I'm not sure whether here $\mathbb{Q}$ is all rational numbers ). And let ...
1
vote
1answer
36 views

Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and ...
0
votes
1answer
20 views

$\lbrace \lim f_n(x) \rbrace$ is a Borel set if each $f_n$ is borel

Suppose for all $n$ that $f_n:\mathbb{R}\to \mathbb{R}$ is Borel measurable. What follows is an attempt of the proof that $\lbrace x: \lim_{n\to \infty} f_n\rbrace$ is Borel measurable, but I am a bit ...
0
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1answer
30 views

Book on probability theory with sigma algebra

Please suggest or recommend a book on Probability theory emphasising on sigma algebra and with Kolmogorov’s axiomatic development.
3
votes
2answers
51 views

from Carathéodory Derivative definition to the derivative of $\sin(x)$

A function $f$ is Carathéodory differentiable at $a$ if there exists a function $\phi$ which is continuous at a such that $$f(x)-f(a)=\phi(x)(x-a).$$ For $f(x) = x^n$, $\phi(x) = x^{n-1} + ...
2
votes
0answers
18 views

Product measure and splitting integral

Let $(X, A, \mu), (Y, B, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu)$, $g \in \mathcal L^1 (\nu)$. I want to show that $fg \in \mathcal L^1 (\mu \otimes \nu)$ $\int fg \ d(\mu \otimes ...
1
vote
1answer
15 views

Measure of a section is zero imply other section will be zero?

How do I see this? Consider two $\sigma$-finite measure spaces $(X,A,\mu)$ and $(Y,B,\lambda)$. And let $E \in A \otimes B$ such that $\mu(E^y)=0$ $\lambda-$a.e. Than $\lambda(E_x)=0$ $\mu$-a.e. ...
0
votes
1answer
11 views

Obtaining Essential Range and Support of a Measurable Function from Estimate

The following is an old real analysis qual problem which I cannot solve. Problem. Let $f\geq 0$ be a measurable function on $\mathbb{R}^{n}$. Suppose there exists $C>0$ such that for all ...
2
votes
2answers
67 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
1
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0answers
13 views

Nonnegative function as limit of monotone increasing sequence (Measure Theory)

I am reading Bartle's "The Elements of Integration" and am at the part where he proves Lemma 2.11: If $f$ is a nonnegative function in $M(X,X)$, then there exists a sequence $(\phi_n )$ in $M(X,X)$ ...
1
vote
1answer
14 views

Continuous random variables and probability density function

OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called ...
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0answers
8 views

Entropy of a 2D function versus 1D function.

I am a novice in information theory so this is more of a question seeking pointers to ideas/references to think further on the thought. I want to make concrete the idea that a function of two ...
0
votes
1answer
21 views

Set of positive measures and Banach space

In measure theory i heard recently a statement in my class, which says that the set of all (positive) measures does not make a Banach space ( whereas the set of signed measures makes up a Banach space ...
2
votes
1answer
29 views

Integration by parts formula with Lebesgue Integral and distribution function

I'm struggling to find a solution for the following problem: Let $f$ be an absolutely continuos function on [a,b], let $\mu$ be a bounded Borel measure on [a,b], and let $\Phi_\mu(t)=\mu([a,t))$ with ...
-1
votes
1answer
43 views

Why do these kind of $f(n)$'s make the limsup statement hold?

Suppose we have a function $g: \mathbb{N} \to \mathbb{N}$ s.t. $g(n) \to \infty$ as $n \to \infty$. Is it true that $g \in \{f \ | \ \limsup A_{f(n)} \subseteq \limsup A_n\}$? Suppose $g \to K$ as ...
0
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0answers
36 views

functions which are almost continuous.

Lusin's Theorem: Suppose $f$ is measurable and finite valued on $E$ with $E$ of finite measure. Then for every $\epsilon >0$ there exists a set $F_{\epsilon}$,with $$F_{\epsilon} \subset E, ...
0
votes
3answers
56 views

Calculate $\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}$

I have tried several methods but even I can not calculate. $$\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}\,dx$$ If anyone can help, it was part of a test and still I ...
0
votes
1answer
13 views

The image of Borel set under measurable mapping

Let $f:\mathbb R^n \to \mathbb R^n$ be a measurable mapping (assuming Lebesgue measure). What we could say about an image of Borel set $B$. Is the set $f(B)$ measurable?
1
vote
1answer
21 views

Properties of Stochastic Interval

I'm reading "Limit Thoerem for Stochastic Processes" and finding it hard to calculate the Stochastic interval.For example : In proposition 2.10,$T$ is a stopping time: If $A\in\mathcal F_0$,I need ...
1
vote
1answer
30 views

Determining if $\int f_n\to 0$ implies that $f_n\to 0$ in measure and $f_n(x)\to 0$ a.e.

If $f_n$ is a sequence of measurable functions on $(X,\mu)$ into $[0,1]$ and $\int f_n\to 0$, I am trying to prove (or disprove the following): (i) $f_n$ converges to $0$ in measure. (ii) For almost ...
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0answers
26 views

How can I show the equivalent condition for $\mu^*$-measurability

This is Exercise 4.15 from "Real Analysis for Graduate Students": Let $X$ be a set and $A$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a pre-measure on $A$ such that ...
1
vote
1answer
28 views

Problems in the integration limits to apply Fubini's theorem

If $f:(0,a)\rightarrow\mathbb{R}$ integrable function and $$g(x)=\int_{x}^a \dfrac{f(t)}{t}dt.$$ Then $g$ is integrable and $\int_{0}^a g(t)dt=\int_{0}^a f(t)dt$. I have to use Fubini's theorem but ...
0
votes
0answers
11 views

Intersection of measurable correspondences

Let $X$ be a separable metrizable space and $(S, \Sigma)$ a measurable space. Let also $\Psi _{1}: S \twoheadrightarrow X$ be a weakly measurable correspondence with nonempty compact values and $\Psi ...
0
votes
1answer
20 views

Is the product of a Schwartz function and a locally integrable function integrable?

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$
1
vote
1answer
64 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
3
votes
1answer
30 views

Convergence for every measurable set

Let $(f_n)$ non-negative measurable functions such that $f_n\to f$ and $\int f_n\to \int f<\infty$. We have to prove that $\int_E f_n\to \int_Ef$ for each $E$ measurable. I know that if $f_n\to f$ ...
1
vote
1answer
41 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
0
votes
2answers
21 views

Predual of $l^1(\Gamma)$

Let $\Gamma$ be an uncountable index set. For example $\Gamma=\mathbb R$. Let $l^1(\Gamma)$ be the set of functions with countable support and finite sum: $$ \sum_{a\in\Gamma}|f(a)|<\infty. $$ The ...
2
votes
0answers
8 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
3
votes
0answers
32 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...
2
votes
1answer
31 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
1
vote
1answer
39 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
0
votes
0answers
29 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
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vote
0answers
10 views

Proof of Kolmogorov zero-one law in measure-theoretic setting

I have met, in some paper, the following form of the Kolmogorov zero-one law used: If $A\subseteq 2^\Bbb N$ is a subset of Cantor space such that when $x,y\in 2^\Bbb N$ are such that $x,y$ differ ...
2
votes
1answer
25 views

Total variation distance is complete

For a given measurable space $X$, $\mathcal{P}(X)$ denotes the space of all the probability measures on $X$. The total variation distance $\rho$ on $\mathcal{P}(X)$ is defined by: for $\mu, \nu \in ...
0
votes
1answer
75 views

Question about Measure Theory

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...