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

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2
votes
1answer
275 views

R as a union of a zero measure set and a meager set

Let $ \left\{ {r_i } \right\}_{i = 1}^\infty = \mathbb{Q}$ an enumeration of $\mathbb{Q}$. Let $ J_{n,i} = \left( {r_i - \frac{1} {{2^{n + i} }},r_i + \frac{1} {{2^{n + i} }}} \right) $ $ ...
2
votes
2answers
252 views

How to show that $\frac{f}{g}$ is measurable

Here is my attempt to show that $\frac{f}{g}~,g\neq 0$ is a measurable function, if $f$ and $g$ are measurable function. I'd be happy if someone could look if it's okay. Since $fg$ is measurable, ...
2
votes
1answer
280 views

Separability of the set of positive measures

Let $X$ be a locally compact separable & metrizable space, and $M^{+}(X)$ its space of positive measures (i.e. positive linear forms on the space of continuous functions on $X$, continuous on each ...
2
votes
1answer
107 views

Jordan Measures without $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$?

I am trying to prove that Jordan measures satisfy with the following properties $A, B \subset \mathbb R$ and $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$, similarly for $B$: $$\bar{\mu} (A) \...
2
votes
1answer
277 views

An example of Markov-Feller chain with some properties

Let $X$ be a Polish space and $C(X)$ denote the space of all bounded and continuous functions on $X$. We consider a Markov chain $(\xi_n)_{n\geq 0}$ with transition probability $P:X\times \mathcal{B}...
2
votes
1answer
136 views

A sequence of functions $f_n \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$

Consider a sequence of functions $\{f_n \}\in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ , convergent to $f$ in $L^1(\mathbb{R})$ and to $g$ in $L^2(\mathbb{R})$. Prove that $f=g$ a.e. What I understood ...
2
votes
1answer
275 views

An exercise for weak convergence

Recently, I found an exercise in Hunter's Applied Analysis(last page in the link), which may be closely related to the question I raised two months ago. Consider heat flow in a rod with rapidly ...
2
votes
1answer
98 views

Random variable problem

Define a discrete random variable. Let $(Ω, A, P )$ be a probability space with $Ω = \{1,2,3,4,5,6\}$ and $F = \{Φ, \{1,3,5\}, \{2,4,6\}, Ω\}$. Define functions $X$, $Y$, $Z$ on $Ω$ as $X(k)= k$, $...
2
votes
1answer
257 views

Show that this function is not continuous except on a set of measure zero

Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals Let $$g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$$ where $$\chi_{(0,1]} = \left\{\begin{array}{ll} 1&...
2
votes
1answer
47 views

Measurable function and the Mean Value Theorem

Let $\,f:[a,b]\to \mathbb{R}\,$ be continuous on $[a,b]$ and derivable on $(a,b)$. By the mean value property, for all $\,x\in (a,b)\,$ there exists $\,\xi_x\in (a,x)\,$ such that $\,f(x)-f(a)=f'\left(...
2
votes
1answer
51 views

$m_*(E)=m^*(E)\iff E$ Lebesgue measurable

Let $E\subset [a,b]$. Show that $E$ is Lebesgue measurable if and only if the Lebesgue outer measure of $E$ is equal to the Lebesgue inner measure of $E$. I have seen the proof for this above ...
2
votes
2answers
77 views

$L^p \subset L^q$

Let $(X,M,\mu)$ be a measure space. Let $\Omega \subset X$ be a measurable set. We have $L^2(\Omega) \subset L^1(\Omega)$ . Can we have that $\mu(\Omega)< \infty $ ?
2
votes
1answer
39 views

Can anyone explain the connection between reverse fatou's lemma and Fatou's lemma?

Here is the version of reverse Fatou's lemma I am looking at. $E_n$ is a sequence of events. $P(\limsup E_n) \geq \limsup P(E_n)$ Here is Fatou's lemma. Let $f_1,f_2,\ldots$ be a sequence ...
2
votes
0answers
26 views

Show linearity of a functional if it holds for nonnegatives

Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\...
2
votes
1answer
41 views

Capacity of a set in $\mathbb{R}^n$

The $2$-capacity of a set $\Omega$ sitting inside an open set $V \subset \mathbb{R}^n$ is given by $$\text{cap}_2(\Omega, V) = \inf_{u \in C^\infty_0(V), u|_\Omega \equiv 1} \int_V |\nabla u|^2 dx.$$ ...
2
votes
0answers
33 views

Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
2
votes
1answer
61 views

Uniqueness of the uniform spherical distribution

Suppose that $X,Y$ are random vectors on some (possibly different) probability spaces mapping to $\mathbb R^n$ for some $n\in\mathbb N$. Suppose furthermore that $\|X\|=r>0$ for all realizations ...
2
votes
1answer
44 views

Another question about proving Lebesgue Decomposition

Note: This is my original question. I have been kindly helped to turn this into a correct proof, which I have posted as an answer so this question won't show up as "unanswered". As an exercise, I am ...
2
votes
0answers
34 views

Monotove Convergence theorem for decreasing sequence

Suppose $f_n: X\to [0, \infty]$ is measurable for $n = 1, 2, 3, ...,$ $f_1 \geqslant f_2 \geqslant f_3 \geqslant · · · \geqslant 0,$ $f_n(x) \to f(x)$ as $n\to \infty$, for every $x\in X$, and $f_1 \...
2
votes
0answers
40 views

Simple proof of uniqueness of Lebesgue Decomposition?

Lebesgue's Decomposition Thm states: if $\lambda,\mu$ are $\sigma$-finite measures on a measurable space $(X,\textbf{X})$, then $\exists$ unique measures $\lambda_1,\lambda_2$ on $(X,\textbf{X})$ s.t. ...
2
votes
0answers
38 views

Integration w. r. t. counting measure

I'm learning about measure theory, specifically integration w.r.t. counting measure, and need help to verify my understanding of this new notion through two exercises. (1) Let $(\mathbb{N},\scr{P}...
2
votes
0answers
22 views

Probability that there exists $M>0$ such that two processes $\{X_t\}$ and $\{Y_t\}$ are smaller than $M$ at the same time, for infinitely many $t$.

Suppose I have two iid sequences of random variables $\{X_t\}_{t\in\mathbb{N}}$ and $\{Y_t\}_{t\in\mathbb{N}}$, both absolutely continuous with full support. I know that with probability one, for any $...
2
votes
0answers
65 views

Is there always a Minimal Product Measure

I am studying measure theory and I have a question concerning the wikipedia-article "Product measure". I already asked on the Wikipedia-"talk"-page but so far noone answered. The problem concerns the "...
2
votes
0answers
22 views

Problem 1.3 from RCA Rudin

Prove that if $f$ is a real function on a measurable space $X$ such that $\{x : f(x) \geqslant r\}$ is measurable for every rational $r$, then $f$ is measurable. Proof: For every $\alpha\in \mathbb{R}...
2
votes
0answers
38 views

Theorem 1.41 Rudin RCA

After reading this theorem I have one question: Using theorem 1.27 we show that $$\int \limits_{X}gd\mu=(1)<\infty.$$ Also we must show that $g$ is measurable to conclude that $g\in L^1(\mu)$. ...
2
votes
0answers
29 views

If $f\in L^1(\mu)$ and $\int \limits_{E}fd\mu=0$ for every $E\in \mathfrak{M}$. Then $f=0$ a.e. on $X$

Suppose $f\in L^1(\mu)$ and $\int \limits_{E}fd\mu=0$ for every $E\in \mathfrak{M}$. Then $f=0$ a.e. on $X$. Proof: First we assume that our function $f$ is real. We have to show that the set $\{x\in ...
2
votes
0answers
27 views

Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
2
votes
0answers
43 views

How can higher dimension spaces have smaller unit balls? [duplicate]

I have recently been shown the gamma function and a few of its uses, and one of those is calculating the measure of the unit ball in $\Bbb{R}^n$. The formula shows the measure going to zero (rather ...
2
votes
0answers
27 views

Why is the Newton quotient measurable when the conditions are like the following.

Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions: for each $x$, $f(x, y)$ is an integrable function of $y$. $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
2
votes
2answers
31 views

Random variable independent of $\sigma$-algebra and conditional expectation

What does it mean to say that a random variable is independent of a sigma-algebra, and why then does this imply that $E(RV| \sigma) = RV$?. I have no clue what this independence stuff is about (...
2
votes
0answers
19 views

$X_t$ measurable wrt $\sigma$-algebra and “revaled information”

Studying stochastic processes, it is mentioned that if $(X)_t$ is a process and $(\mathcal{X})_t$ a filtration, then if the process is adapted to the filtration, the informal way to think about it is ...
2
votes
0answers
29 views

A detail on Fubini's theorem

Let $f(x, y)$ be a measurable function on a product of two balls $B_{1}$ and $B_{2}$ in $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$ respectively and $m,n\geq1$. We know, according to Fubini's theorem, that ...
2
votes
0answers
22 views

What is a good way to think of Caratheodory Hahn theorem intuitively?

I am reading the theorem in Royden's book on Caratheodory Hahn theorem. After reading it for like 10 times, I still do not quite understand it. Can anyone please offer some intuition?
2
votes
2answers
71 views

Sum of measurable functions is measurable: countable choice required?

The standard proof that the sum of measurable functions is measurable uses countable choice, via the countable subadditivity of outer measure ($\implies$ measurable sets are closed under countable ...
2
votes
1answer
44 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces $(\Omega^i,\mathcal{F}^...
2
votes
0answers
38 views

Prerequisites to understanding proof of Fubini's Theorem? [closed]

I'm currently studying tensor analysis, and I have studied elementary calculus (meaning calc I, II, III, and diffy Q), as well as linear algebra. Given all of this, what are the rest of the required ...
2
votes
1answer
31 views

Can anyone explain the connections among ring, semiring, algebra, sigmaalgebra in the scope of measure theory?

Can anyone explain the connections among ring, semiring, algebra, sigma-algebra in the scope of measure theory and why are these concepts important?
2
votes
0answers
24 views

Why are the positive measure and negative measure induced by the Hahn Decomposition mutually singular?

The following statement describes the Hahn decomposition and claims that the induced positive measure and negative measure are mutually singular. Why is that the case? On a separate note, what are ...
2
votes
2answers
31 views

Exterior measure does not satisfy additivity. [closed]

Exterior measure $m^*$ satisfy subadditivity. $$m^*(A\cup B) \leq m^*(A) + m^*(B).$$ But for disjoint sets $A$ and $B$ it may be that $m^*(A\cup B) < m^*(A)+m^*(B)$. So I learned that It is one ...
2
votes
0answers
36 views

The relationship between random variables, distribution functions and probability measures

Given a probability space $(\Omega,\mathcal{F},P)$, and a random variable $X\colon\Omega\to\Bbb{R}$, we can associate with it its distribution function $F\colon \Bbb{R}\to[0,1]$ defined as \begin{...
2
votes
1answer
50 views

$\int_{A} fdxdy=0 $ For every rectangle A with area 1. Then is it f=0 a.e? [closed]

Is it true that the function $f \colon \mathbb R^2 \to \mathbb R$ satisfying condition $\int_{A} f \,\textrm{d}x \,\textrm{d}y=0$ for every rectangle $A$ whose area is $1$ must be identically 0 ...
2
votes
0answers
32 views

Prove that if $f$ is measurable, then $f(Tx)$ is measurable.

Definition of measurability of function $f$ is said to be measurable if $\{x:f(x)>a\}$ is measurable. Prove that, for $f$ defined and measurable in $\mathbb{R}^n$, $f(Tx)$ is measurable, where $...
2
votes
0answers
39 views

Lebesgue measure for sup groups of $\mathbb R$

I want to ask about Lebesgue measure for groups I know $m^*(\mathbb N)=0$ and $m^*(\mathbb Q)=0$ $m^*(\mathbb R)=infinite $ $m^*(${$0,1$}$)=0$ $\mathbb N$ is a normal numbers, $\mathbb Q$ is a ...
2
votes
0answers
34 views

Existence of a measurable function

The question is Let $\theta>0$ fixed. Is there a measurable function $h:[0,1]\to\mathbb{R}$ such that $\int_0^1h(x)x^{\theta-1}dx=1$ and $h$ is independent of $\theta$? Could you check my ...
2
votes
0answers
39 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's "Probability and Measure" (and other books) show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say the joint measurability ...
2
votes
0answers
88 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
2
votes
1answer
30 views

show that continuous functions on $\mathbb{R}$ are measurable

I am trying to show this using the theorem: A function $f: \Omega \to \mathbb{R}$ is measurable if and only if $f^{-1}(E) \in \mathcal{F}$ for all borel sets $E$. The proof to show a continuous ...
2
votes
0answers
37 views

f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel (or at least Lebesgue Measurable)?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...