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

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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 ...
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1answer
11 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) = ...
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0answers
9 views

Weak convergence and subsequences of subsequences of distribution functions

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 $\mu$ ...
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1answer
8 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 $\mathbb{Q} = \otimes_{i=1}^n \mathbb{Q_i}$ ...
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1answer
14 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$ ...
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0answers
13 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 ...
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0answers
12 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 ...
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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
29 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 | ...
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0answers
18 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 $F_{\mu_n}(t):=\mu_n([a,t))$ ...
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2answers
64 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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2answers
45 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
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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 ...
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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. ...
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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 ...
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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 ...
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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 $M(X,X)$ ...
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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 ...
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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 ...
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1answer
24 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 ...
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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 ...
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0answers
35 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, ...
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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 ...
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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?
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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 ...
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1answer
29 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 ...
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1answer
25 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 ...
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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 ...
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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 $$ \int |fg| <\infty. $$
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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 ...
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1answer
27 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 $f_n\to f$ ...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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0answers
28 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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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 ...
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1answer
70 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 ...
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0answers
19 views

measure equality on two sigma algebras also holds on the combined sigma algebra?

we have the following setup: $Q, P \text{ are measures on the }\sigma-\text{algebras } \mathcal{F} \text{ and } \mathcal{G} $. Let $P(A)=Q(A) \forall A\in\mathcal{F}$ and $P(B)=Q(B) \forall ...