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

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14 views

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) ...
2
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1answer
22 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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1answer
18 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.

$\mu(E)\ge \nu(E)\ \forall E\in A\ \Rightarrow\ \mu(\cup E)\ge\nu(\cup E)$? Here $\mu,\nu$ are probability measures on a $\sigma$-algebra $M$ on a set $X$. We can assume that arbitrary unions of the ...
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1answer
44 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
8 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 ...
2
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2answers
42 views

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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0answers
40 views

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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0answers
13 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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1answer
27 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
33 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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0answers
22 views

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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0answers
21 views

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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0answers
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
70 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 ...
2
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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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2answers
74 views

$\lim_{n \to \infty} \int_{0}^{n}(1-\frac{3x}{n})^ne^{\frac{x}{2}}dx$=?

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

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 ...
2
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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
10 views

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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3answers
105 views

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 ...
0
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1answer
20 views

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 ...
1
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3answers
46 views

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
8 views

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: ...
1
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1answer
13 views

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. ...
0
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1answer
27 views

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
29 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
26 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 $A_{1}$ ...
4
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2answers
69 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 ...
2
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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 ...
5
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1answer
42 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 ...
2
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0answers
26 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 ...
0
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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 ...
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0answers
19 views

Proving that if $f\in\mathcal{F}C^{1}_{b}(X)$ then $f\in W^{1,p}(X,\gamma_{1})$ for $p>1$

Let $X$ be a separable Banach space endowed with a centered nondegenerate Gaussian measure $\gamma$. Then consider $f\in\mathcal{F}C^{1}_{b}(X)$. I want to prove that $f$ is in ...
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0answers
23 views

Total measure and Riesz theorem

As I specified in the other question I asked, analysis is not my field, so I'm sorry if my question is trivial. Riesz representation theorem states the following: Given $X$ a locally compact ...
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0answers
9 views

“Property” preserving property under $\sigma-$generating classes

Let $\Omega$ be a non-empty set, $\mathcal{F}$ be a $\sigma$-algebra of subsets of $\Omega$ and $\mathcal{C}$ be a class of subsets of $\Omega$ such that $\sigma(\mathcal{C})=\mathcal{F}$. If ...
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1answer
16 views

The collection of open intervals is not a $\sigma$-algebra

We need to show that the collection of open intervals is not a $\sigma$-algebra. For it to be a $\sigma$-algebra, we need to show that $\emptyset, \Omega$ is in the $\sigma$-algebra, which it is. ...
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0answers
16 views

Is the following construction a measure?

Please excuse my informality, but I am currently working on the intuition behind the idea. Suppose two parameters are given: An event $E$ composed of basic events, like union and intersection of ...
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0answers
18 views

Does $\sigma$-additivity imply continuity?

Given a measurable space $(X, \mathcal A)$ and a $\sigma$-additive function $\mu$ on $A$, is $\mu$ continuous from below? I.e. is the following condition satisfied? If $(A_n)_n \subset \mathcal A$ is ...
1
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0answers
14 views

Linear transformations and Lebesgue measure

Let $d,d'\in\mathbb{N}$, $d'<d$, and for all $i\in\{1,\ldots,d\}$ let $A_i\in\mathbb{R}^{(d-1)\times d}$ be the matrix one gets if he cancels the $i$-th row of the $d$-dimensional identity matrix, ...
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0answers
13 views

Showing the bijection of a function on the set of finite measures on a singleton

Let the set $E = \{a\}$ and $M(E) = \{\mu : \mathcal{B}(E) \to \mathbb{R}_+ : \mu$ finite measure$\}$. Consider the function $f : M(E) \to \mathbb{R}_+$, defined as follows: $$f(\mu) = ...
4
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2answers
66 views

Prove that if $X$ is subgaussian, then ${\bf E}e^{tX}=1+\sum_{k=1}^{\infty}\frac{t^k}{k!}{\bf E}X^k$

Prove that if $X$ is subgaussian, then $${\bf E}e^{tX}=1+\sum_{k=1}^{\infty}\frac{t^k}{k!}{\bf E}X^k$$ So basically I just need to push the integral through the infinite sum $${\bf ...
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vote
2answers
39 views

Example where integration by parts formula fails for a.e. differentiable functions

I'm studying for a qual and found this problem. We were given two absolutely continuous functions $f,g$ on $[a,b]$. The first two parts of the problem involved proving the integration by parts ...
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1answer
30 views

Show that $\mathcal{O}$ forms a $\sigma$-algebra

Let $\Omega$ any uncountable set and $\mathcal{O}$ is the collection of all subsets of $\Omega$ which are countable or have countable complements be the collection. We want to show that $\mathcal{O}$ ...
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0answers
16 views

Show that $\mathcal{O}$ is a $\sigma$-algebra that contains $\mathcal{C}$.

Let $\mathcal{C}$ be any nonempty family of subsets of $\Omega$. Consider the collection of all $\sigma$-algebras that contain $\mathcal{C}$ ($\mathcal{C}\subset 2^\Omega$ for example). Let ...
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1answer
15 views

How to understand an $S$-valued random variable (Simple example)

How to understand an $S$-valued random variable (in a stochastic process) and how is filtration adapted to that variable? How can you give a simple example of defined probability measure and random ...
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0answers
18 views

Product sigma algebra in the countable case

Let $(\Omega_i,\mathcal{F_i})_{i \in \mathbb{N}}$ be the countable family of measurable spaces.Then I was wondering whether the canonical product sigma algebra on the product space that is for example ...
0
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1answer
22 views

a function which does not converge in $L^2$

Please, help me, with this issue: Let $f_n:(0,1]\to \mathbb{R}$, $f_n(x)=\frac{n}{1+n\sqrt x}$. I was asked to show that $f_n\in L^2(0,1]$. I have to prove that $$\left( \int_{(0,1]} f_n(x)^2~d\lambda ...
0
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0answers
22 views

Borel functions propreties?

let $(X,\beta,m)$ a measurable space and $m$ is a $\sigma$-finite measure, and $h:X\rightarrow \mathbb C$ a Borel function ($\beta$-measurable) such that $h\in L^2(X)$. Let $X_n=h^{-1}[-n,n]$ so can ...