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

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Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...
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If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable?

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable? I am not sure here if the limsup condition here is as strong as if I ...
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1answer
24 views

Is there a measure which allows me to tell how closely something is to an ellipse?

Roundness is the measure of how closely the shape of an object approaches that of a circle. I am trying to find a similar measure which shows how closely is something to an ellipse. Is there any ...
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11 views

Counting measure on sigma algebra power set of natural numbers .

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
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1answer
26 views

Lebesgue integral of vector-valued function?

In Bernt Øksendals stochastic differential equations he says that if we have a random variable $X:\Omega\rightarrow\mathbb{R}^d$. He defines the expectation: $E[X]=\int_\Omega ...
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1answer
17 views

Congruent measurable sets

I have a question regarding Congruent relations: In Euclidean geometry, two subsets of $\mathbb{R}^{d}$ are said to be congruent if one set can be mapped onto the other by translations and rotations. ...
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1answer
21 views

Weak convergence in $L^p$ equivalent to pointwise almost everywhere convergence

Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence? Let us also assume the measure space is ...
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1answer
24 views

Equality in Conditional Jensen's Inequality

Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid ...
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1answer
20 views

Proving measurability in $\mathbb{R}^2$

I am given the problem: suppose for measurable, real-valued functions $f$ and $g$, and an open set $A \subset \mathbb R ^2$, prove that $\{x \in \mathbb R : (f(x),g(x)) \in A\}$ is a measurable set. ...
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Getting conditional expectation using the Radon-Nikodym derivative for multidimensional random variables. [duplicate]

This is from this Wikipedia article: https://en.wikipedia.org/wiki/Conditional_expectation#Conditional_expectation_with_respect_to_a_.CF.83-algebra We have a probabilitys pace $(\Omega, \mathcal{F}, ...
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$g(x) = sup_{α∈A} (f_α(x))$, $x ∈ E$ need not be a measurable function.

We know that if $(f_n)$ is a sequence of measurable functions on $E$, then $g = sup_n f_n$ defined as $g(x) = sup f_n(x)$, $x ∈ E_ n$ is a measurable function. Prove by an example that if $A$ is an ...
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1answer
38 views

Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$.

Let $I = [0,1]$ and $χ_I : \Bbb R → \Bbb R$ be the characteristic function on $I$. Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$. ...
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1answer
19 views

examples of random variables that the result of their preimage is not in F?

let's assume we have a probability space $(\Omega , F , P)$. and we have a random variable $X$ defined as : $X : \Omega \rightarrow \Bbb{R}$ and we also use a Borel set ($\mathcal{B}$).(making the ...
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1answer
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Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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22 views

Exercise in “Elements of Integration” by Bartle

I found the problem below in Bartle's book "The Elements of Integration and Lebesgue Measure". I have not been able to solve it. All ideas are welcome. If $\phi$ is not uniformly continuous, then ...
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19 views

Weak convergence of measures and compact sets

Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I ...
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Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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1answer
41 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
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27 views

$\mu(E)\ge \nu(E)\ \forall E\in A\ \Rightarrow\ \mu(\cup E)\ge\nu(\cup E)$? Here $\mu,\nu$ are probability measures on a $\sigma$-algebra.

$\mu(E)\ge \nu(E)\ \forall E\in A\ \Rightarrow\ \mu(\cup E)\ge\nu(\cup E)$? A. Here $\mu,\nu$ are probability measures on a $\sigma$-algebra $M$ on a set $X$. We can assume that arbitrary unions of ...
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49 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
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1answer
12 views

Partition generated $\sigma$-algebra

I saw this example given as a $\sigma$-algebra in various places. It goes like this: Let $X$ be a set and assume that the collection $\{A_1,\dots, A_N\}$ is a partition of $X$. Then the collection ...
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Does the following condition implies full outer measure?

Let $X \subseteq 2^{\omega}$ be a set of positive Lebesgue measure. Suppose that for every $\eta, \nu \in 2^{<\omega}$ of the same length, the measure of $X$ above $\eta$ is the same as the measure ...
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Concavity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Is the function $f(r) = \mu(A_r)^{1/n}$ ...
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16 views

Lebesgue measure of region under curve

Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Theorem: Define the area under the graph of $f$ to be ...
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34 views

Suppose $X_n \to_{p} X$, if $\limsup_n E|X_n|^r \leq E|X|^r$, how can I show that $X_n \to_r X$?

If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to ...
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1answer
34 views

If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$?

If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$? The problem is the following: In the solution of Folland chapter 1 exercise 14, ...
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why is the collection of all finite subsets of $\mathbb{R}$ not a $\sigma-ring$

It says the definition of a $\sigma-ring$ is if $A,B \in \mathcal R$ then $A \setminus B \in \mathcal R$ and if $ A_{n} \in \mathcal R \forall n \in \mathbb{N}$ then $\cup_{1}^{\infty}A_{n} \in ...
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Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
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1answer
28 views

A doubt on a proof of a theorem of Durret's Probability Theory

Below is the text of the theorem: $\mathcal{F}_{i,j}$ are sigma algebras indexed by $i$ and $j$. I'm having some difficulties in understanding this proof. Do the $\mathcal{A}_i$ contain $\Omega$ ...
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19 views

Integral inequality $L^p$ spaces

I'm trying to solve this problem: Let $1<p<\infty$. Then let $f:(0,\infty)\to [0,\infty]$ a measurable non negative function. It's true the following inequality: $$\int_0^\infty ( ...
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1answer
72 views

$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=$?

$$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=?$$ I am allowed to used all the classical techniques of calculus, and this was a question from measure theory when we were ...
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1answer
34 views

Disentangling $\int_Af(\mathbf{x})\ d\mathbf{x}$, using Fubini Theorem.

Let $\mathcal{B}^n$ be the borel sigma algebra generated by the rectangles in $\mathbb{R}^n$. I can write $f(\mathbf{x})=g_1(x_1)\cdots g_n(x_n)$. Let $\mu=\mu_1\times \cdots \times \mu_n$ be the ...
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$\lim_{n \to \infty} \int_{0}^{n}(1-\frac{3x}{n})^ne^{\frac{x}{2}}dx$=?

$$\lim_{n \to \infty} \int_{0}^{n}\left(1-\frac{3x}{n}\right)^ne^{\frac{x}{2}}dx$$ I thought about using the theorem of monotonic convergence and had ...
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Integrals of functions of statistics

Let $X: \Omega \to \mathbb{R}^n$ be a measurable random vector with law $\Lambda_X$ and probability density function (pdf) $f_X$. Let $T:\mathbb{R}^n \to \mathbb{R}^2$ a statistic (a ...
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1answer
30 views

All measures $\alpha,\beta$ on $[0,1]$ satisfying certain moment conditions

This is a problem I found trying to find some properties related to exchangeable sequences. Anyway, I am not able to find a characterization of all solutions.. I know there are at least two completely ...
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1answer
11 views

Countability of generated ring $R(E)$

I am studying Paul R. Halmos Measure theory. In the section 5 of chapter 1, theorem 5 states that : If $E$ is a countable class of sets, then $R(E)$ is countable. The proof uses class of all finite ...
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1answer
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How to turn convergence in probability a statement involving n?

Def: for every $\epsilon$ $\lim_{n}P(|X_n-X|>\epsilon)=0$ How to turn it into a statement of there is an N s.t. n>N... Shall we make it $P(\lim_{n}|X_n-X|>\epsilon)=0$ first?
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Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.

Prove $\displaystyle\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}\,dx$ exists. I want to use Dominated convergence theorem to show the ...
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Adapted and progressive processes

Could you please help me proving rigorously the following fact from Mayer's book: (a) if $X_t$ is a process adapted with respect to filtration $\{\mathcal{F}_t\}_{t\ge 0}$ and for every ...
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3answers
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If $\mu(X) < \infty$, $f_n \to f$ a.e., and $\int f_n^2 \leq C$, then $f_n \to f$ in $L^1$

I should be able to get this problem...I'm studying for a qualifying exam and the question is to show that if $\mu(X) < \infty$, $f_n \to f$ a.e., and $\int f_n^2 \leq C$, then $f_n \to f$ in ...
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0answers
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Algebra generated by point cylinders

Let $X\equiv\mathbb N^{\mathbb N}$ denote the set of all sequences of positive integers. For a fixed $n\in\mathbb N$ and $(y_1,\ldots,y_n)\in\mathbb N^n$, define the “point cylinder” as follows: ...
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1answer
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Bump functions converging to an indicator

Suppose $K\subset\mathbb{R}^n$ has a smooth boundary, and let $\phi_s(x)$ be bump functions converging pointwise to the indicator of $K$, i.e. ...
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What is an example of $f \in L^1(\mathbf{R})$ such that $\sum_{n=1}^\infty f(nx)$ converges a.e. but is not in $L^1(\mathbf{R})$

what is an example of $f \in L^1(\mathbf{R})$ such that $\sum_{n=1}^\infty f(nx)$ converges a.e. but is not in $L^1(\mathbf{R})$? Context: This question appeared on an old qualifying exam. I tried ...
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2answers
31 views

Further generalising Holder's inequality

I have proved the following theorem in an earlier part of the question: Let $p,q \geq 1$ be such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that: $$\|fg\|_1 \leq \|f\|_p \|g\|_q$$. I proved this ...
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1answer
27 views

Find the smallest ring generated by all singleton subsets of an uncountable set.

Let $X$ be an uncountable set and let $E$ be the collection of all singleton subsets of $X$. Find the smallest ring generated by $E$. Attempt: Let $R$ be the ring generated by $E$. Let $A_{1}$ ...
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72 views

$f_n \geq 0$ and $\int f_n = 1$ implies $\limsup_n \left( f_n(x) \right)^{\frac{1}{n}} \leq 1$ for a.e. $x$

I am studying for a qualifying exam and am having difficulty with this problem: Let $\left( X, \mathcal{M}, \mu \right)$ be a measure space and assume $f_n \geq 0$ such that $\int f_n = 1$ for all ...
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1answer
27 views

$X,Y$ are independent and $f(X,Y)$ and $Y$ are independent. Does there exists a $g$ s.t. $f(X,Y)=g(X)$ a.s.?

Let $(E,\mathcal{E}),(F,\mathcal{F})$ and $(G,\mathcal{G})$ be measure spaces and $f:E\times F\rightarrow G$ a measurable function. $X,Y$ are independent RV with values in $E$ and $F$. In my ...
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1answer
43 views

Definition of Sigma Algebra

I was wondering, why are we not allowed to take arbitrary unions (likewise intersections) in the definition of a sigma algebra?; I am looking for a more or less intuitive reason. It seems to me that ...
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0answers
29 views

Radon measures and Holder distributions

EDIT: sorry I realized I made some mistakes asking the question, so I'm fixing them. Analysis is not really my field, so I hope this question is not too trivial. Let's consider $X$ a locally compact ...
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1answer
14 views

Is a metric space a requirements for the application of the algebra of events from probability?

When I refer to a metric space, I mean a space that has some genuine notion of distance. In some applied context, this distance would be computed with respect to a coordinate system. I just wanted to ...