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

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Proving the middle thirds cantor set contains interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
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Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
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1answer
17 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
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7 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
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1answer
6 views

A relation between the inner and outer jordan measures

I'm studying measure theory and I was thinking about the following question: Is it true that whenever $A\subset B\subset \mathbb{R}^n$ are bounded, $\qquad \qquad \qquad \qquad \qquad \qquad ...
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1answer
16 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
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13 views

Automophism of G and Haar measure

Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform ...
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1answer
30 views

Measurable set limit

If $\forall n \in ℕ$ , $ f_n: (X,M) \rightarrow (\overline{\mathbb{R}},B) $ are measurable. (where X is any space, M is a sigma algebra, B is Borel sigma algebra) Prove that the set $A = \{x\in X: ...
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1answer
22 views

How to show that the function $g(x)=x|\sin(x^{-1/2})|$ is absolutely continuous?

I am having trouble showing the on $[0,1]$, $g(x):=x\mid\sin(x^{-1/2})\ \mid$ when $x>0$ and $0$ is $x=0$ is absolutely continuous. I was instructed to try: $\ m(A) < \delta \Rightarrow ...
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5answers
104 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
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1answer
22 views

Relation between the modulus of integrability and $L^p$ spaces

Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Given an integrable function $f$ on $X$, we can quantify its integrability in multiple ways. One is the modulus of integrability, which is a ...
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2answers
258 views

Measure - exercise 22 from Folland

I'm doing some exercises from Folland's real analysis book. Exercise 18 is done and should help to do exercise 22, but I'm stuck. The definition of completion is given below. This is not ...
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21 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
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1answer
15 views

Independence of random variables involving Brownian motion

I am reading a book on stochastic analysis and I don't understand the following (i.e. don't know how to prove it rigorously): Let $B$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the ...
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50 views

Why does the union of all open null sets is itself a nullset for second countable space?

On the online Encyclopedia of mathematics, it is written "The existence of a countable base guarantees that the union of all open μ-null sets is itself a nullset." See: ...
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25 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
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0answers
29 views

Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
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2answers
48 views

Probabilistic implications of the existence of non-measurable sets

Measure theory and probability theory are deeply connected through the interpretation of subset measures on the sample space as probabilities of events. A major (and somewhat disturbing) result from ...
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2answers
49 views

If $\int_E f=\int_E g$ then $f=g$ a.e.?

Is the converse of the following statement is true? Let $f$ and $g$ be two bounded measurable functions on a set $E$. If $f(x)=g(x)$ a.e. on $E$ then $$\int_E f=\int_E g$$ Here is my proof for ...
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59 views

Borel measurability is a local property

I am looking at Exercise 5.2 (page 44) in "Real Analysis for Graduate Students" by Richard Bass. Let $f:(0, 1)\to \mathbb{R}$ be a function such that for every $x\in (0, 1)$, there exist ...
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0answers
13 views

Requirements for existence Lebesgue-Stieltjes measure corresponding to distribution function in $\mathbb{R}^n$

I am going through Ash's book "Probability and Measure Theory". It says that: We know that a distribution function of $\mathbb{R}$ determines a corresponding Lebesgue-Stieltjes measure. This is true ...
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1answer
60 views

A question regarding non-(Lebesgue)-measurable sets in models of ZFC+$2^{\aleph_0}$=$\aleph_2$

Let $\mathscr V$ represent a set of Vitali's type. It is known that $|\mathscr V|=2^{\aleph_0}$. Does $\mathscr V$ have any measure-theoretic properties in models of, say, ...
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2answers
30 views

$L^1(X)$, delta epsilon measure proof

Let $f \in L^1(X)$ with $f \ge 0$. We know that $$\nu(E) := \int_E f\,d\mu$$defines a measure on $\Sigma$. How do I show that for every $ \epsilon > 0$ there exists $\delta > 0$ so that for any ...
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1answer
31 views

Formal introduction of product measure

Having proven the existence and uniqueness of the product measure $\mu$, given two $\sigma$-finite measure spaces $(\Omega_1,\sum_1, \mu_1), (\Omega_2,\sum_2, \mu_2)$, you can extend this principle to ...
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39 views

Standard machine in measure theory

Step 1.Prove the property for $h$ which is an indicator function. Step 2.Using linearity, extend the property to all simple positive functions. Step 3. Using Monotone property extend the ...
4
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1answer
51 views

Show that this set, defined similarly to the Cantor set, also has measure 0

The standard, middle-thirds Cantor set can be thought of as the set of all numbers on the interval $[0, 1]$ whose ternary expansions contain no 1s, that is, numbers of the form $$\sum_{n=1}^{\infty} ...
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1answer
32 views

Prove that $\int_X f \, d\mu=\int_Y\mu(f^{-1}[t,\infty)) \, d\mathcal{L}(t)$

Let $\mathcal{L}$ be the Lebesgue measure on $Y=[0,\infty)$. Let $(X,\mathfrak{B},\mu)$ be a $\sigma$-finite measure space and let $f$ be a nonnegative $\mu$-measurable function on $X$. Prove that ...
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3answers
20 views

Indicator in expectation

Suppose we have a measure space $(\Omega,\mathcal{F},P)$, say we have a random variable $X$ defined on this measure space. My question now is; if we have an event say $F \in \mathcal{F}$ is it in ...
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Construction of a strictly increasing, continuous function with zero derivative ae using the Cantor function

It is known that the Cantor function $\psi$ is a non-decreasing, continuous function with zero derivative ae. Let $n,k \in \mathbb N^+$ such that $k < 2^n$, and set: \begin{equation} f_{n,k}(x) = ...
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2answers
36 views

Suppose $P(X \in B) \in \{0,1\}$ for all $B \in \mathcal B(\mathbb R)$. Show $X = c$, $P$-almost-surely.

Let $(\Omega, \mathcal F, \mathcal P)$ be a probability space and let $X$ be a random variable. Suppose $P(X \in B) \in \{0,1\}$ for all $B \in \mathcal B(\mathbb R)$. I want to show that there ...
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positive integrable part implies downside integrable

Let $A: M\rightarrow GL(d)$ measurable where $(M, \mathcal{B},\mu)$ is a probability space, then are equivalent: $$\log^+\Vert A^{\pm1}(x)\Vert\in L^1(\mu)\Leftrightarrow \log^-\Vert ...
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1answer
63 views

Approximate normal distribution

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
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1answer
26 views

Prelim problem in real analysis (any hints)

Find all constants $K > 0$ for which the following holds: If $(X,\Sigma,\mu)$ is any positive measure space and if $f:X\to \mathbb{R} $ is $\mu$ integrable satisfying $\left|\int_E ...
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1answer
22 views

Extension of linear functional on $L^1$

Let $L^1([0,1])$ with the Lebsgue measure. Construct a bounded linear functional on some subspace of some $L^1([0,1])$ which has two distinct norm-preserving linear extensions to $L^1([0,1])$. For ...
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2answers
27 views

Using the MCT to evaluate the integral of a series

I'm studying for my Measure Theory final and I've come across a question that I can't seem to find an answer for. For each $n \in \mathbb{N}$ set $E_n:=[n,2n]$ and let $f:\mathbb{R} \to \mathbb{R}$ ...
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1answer
23 views

Sequence of continuous functions in comple metric space.

Let $\{f_n\}$ be a sequence of continuous complex functions on a (nonempty) complete metric space $X$, such that $f(x)=\displaystyle \lim_{n\to\infty} f_n(x)$ exists for every $x\in X$. a) Prove that ...
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3answers
45 views

Function in $L^\infty$ is element of $L^2$?

Let $\mu$ positive measure and $f\in L^\infty(\mu)$. My question is: $f\in L^2(\mu)$? Thank you all.
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3answers
50 views

Function converging in $L^1$

I've been having trouble with rigorously showing that on the interval $[0, 2\pi] $ the function $\sin^n(nx) \rightarrow 0$ in $L^1$ convergence. I've convinced myself that this is true intuitively by ...
4
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1answer
54 views

What are some practical applications of measure theory apart from providing theoretically rigourous foundations?

It seems that measure theory has a very good theoretical purpose, in that it provides a rigorous framework to define a lot of what we do in analysis. However, I have a hard time thinking of a ...
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1answer
31 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
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1answer
68 views

Convergence in measure implies convergence almost everywhere (on a countable set!)

Here is an interesting problem from "Real Analysis for Graduate Students" by Richard Bass (which is an amazing book, by the way). Suppose $(X, \mathcal{A}, \mu)$ is a measure space, and $X$ is a ...
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Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in ...
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1answer
38 views

Distribution of transformed random variables

We have that f is a density w.r.t the lebesgue measure $m$ for a probability measure on $\mathbb{R}$, that f is continuous and strictly positive. X and Y are to random variables s.t. the distribution ...
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2answers
32 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for ...
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0answers
27 views

Compacticity of distribution functions

I came up with the following assertion and I am having hard time to justify. The author says the following: "Consider a closed and convex set of probability measures on a compact set, say $[0,1]$. ...
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19 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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1answer
48 views

Compute almost sure limit of martingale?

Let $Y_1, Y_2, \dots$ be nonnegative i.i.d random variables with mean 1. Let $$X_n = \prod_{m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim_{n->\infty}X_n = 0$ almost surely. I feel like ...
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1answer
63 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
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0answers
39 views

Lebesgue measure is separable?

I would like to better understand the following definition: $(M, \mathcal {A}, \mu) $ a probability space is separable if there exists a countable family $ \mathcal {E} \subset \mathcal {A} $ such ...
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1answer
77 views

$\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$ implies $\lim_{n \to \infty} \int_B f_n \, d\mu = \int_B f \, d\mu$ for $B \subseteq X$

I'm having trouble with the following problem. Let $(X, \mathcal{M},\mu)$ be a measure space, where $X = [a,b] \subset \mathbb{R}$ is a closed and bounded interval and $\mu$ is the Lebesgue measure. ...