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

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2
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0answers
34 views
+100

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
1
vote
2answers
113 views

Showing independence of a limsup of an independent sequence

Let $\{X_n\}_{n \geq 1}$ be an independent sequence of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Fix $n \geq 1$. I want to prove that $X_1, \ldots, X_n$ is independent of $\limsup X_n$. ...
2
votes
1answer
32 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
0
votes
0answers
15 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
3
votes
1answer
32 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
6
votes
1answer
596 views

Banach Tarski Paradox

I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five ...
1
vote
1answer
31 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
0
votes
1answer
17 views

Probability, approximation by simple functions, boundedness and non-negativity.

In my probability class various results were announced requiring that a certain random variable was bounded. The specific example that interests me is the following: if $Y$ is $\mathcal{G}$-measurable ...
1
vote
1answer
72 views
+50

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...
1
vote
1answer
26 views

Which are the measurable sets with respect to this construction of an outer dirac-type measure?

Let $x \in \Omega$ and $\mathcal{A} \subset\mathcal{P}(X)$ an at most countable set with $\emptyset, \Omega \in \mathcal{A}$. Let $$ \delta_x: \mathcal{A} \to [0,\infty], \quad A \mapsto ...
0
votes
0answers
14 views

On extension of real valued functions from $(0,1)^{\infty}$ onto $[0,1]^{\infty}$

Let $\lambda$ be the standard Haar measure in $[0,1]^{\infty}$. Let $f$ be a bounded real-valued function on $(0,1)^{\infty}$ such that the set of all discontinuity points of $f$ has ...
8
votes
4answers
3k views

Real Analysis Book Choice

I am currently planning to get a book on Real Analysis for self studying before diving into my 4th year real analysis course. The standard textbook for my 4th year course is Stein's Measure, but I ...
0
votes
1answer
27 views

Integration in complex measure

Let $v$ be a complex measure in $(X,M)$. Then $L^{1}(v)=L^{1}(|v|)$. I have made: $L^1(v)\subset L^1(|v|)$?. Let $g\in L^1(v)$ As $v<<|v|$ and $|v|$ is finite measure, then for chain rule, ...
0
votes
0answers
11 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\phi \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K \in ...
3
votes
1answer
29 views

Transformation shift measurable

How to prove that this transformation is measurable? $\sigma:B(n)\rightarrow{B(n)} $ $\sigma(x)(k)=x(k+1)$ $\sigma(...,x_{-1},x_{0},x_{1},...)=(...,x_{0},x_{1},x_{2},...)$ where $B(n)$ with ...
4
votes
1answer
50 views

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<2^{\aleph_0}$, is it true that $X$ is null? Of course I am not assuming the Continuum Hypothesis. EDIT: It might be helpful to know that all ...
2
votes
2answers
18 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
2
votes
3answers
547 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
1
vote
0answers
14 views

Countable unions of Vitali sets…

Let $A \subset \mathbb{R}$ be sets of positive Lebesgue measure. Let $\Gamma$ be a countable dense subgroup of the additive group $\mathbb{R}$. Consider the partition of $\mathbb{R}$ canonically ...
0
votes
2answers
53 views

Radon-Nikodym derivative of Measures [on hold]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia,lecture notes with this blog post by John Baez mentioning it as it ...
-1
votes
1answer
40 views

Proposition on limsup

Given sets (or events) $A_1, A_2, A_3, ...$ and function $f: \mathbb{N} \to \mathbb{N}$, show that $\limsup A_{f(n)} \subseteq \limsup A_n \Leftrightarrow f(n) \to \infty$ This is what @Did told me ...
1
vote
1answer
20 views

Fatou's Lemma conditions for strict inequality

Under what conditions do we have equality (resp. strict inequality) in Fatou's Lemma? If the sequence $f_n$ is convergent, then it is obvious that equality holds. Is it the only case? There are some ...
1
vote
0answers
16 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
6
votes
1answer
1k views

Proving the measure of an increasing sequence of measurable sets is the limit of the measures

Show that if $A_1\subseteq A_2\subseteq A_3\cdots$ is an increasing sequence of measurable sets(so $A_j\subseteq A_{j+1}$ for every positive integer $j$),then we have ...
2
votes
1answer
33 views

Haar measure on locally sigma-compact metric groups

Haar measure on locally sigma-compact metric groups $G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar ...
1
vote
1answer
39 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
2
votes
1answer
58 views

invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
0
votes
0answers
31 views

A Measure Problem on Stein's Real Analysis

I'm considering problem 5 on Stein's real analysis chapter 6 $X$ is a metric space, for any positive linear functional $l$ on $C_0 (X)$ which are the continuous functional on $X$ supported in some ...
2
votes
0answers
36 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
6
votes
1answer
379 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
0answers
25 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
0
votes
1answer
122 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this? I guess $A$ must be an $G_{\delta}$ set which is dense in $\Bbb ...
1
vote
0answers
25 views

Measurable maps in metric spaces.

i have several questions about measurability of maps with values in metric spaces : 1/ When $X$ and $Y$ are two separable metric spaces, it is easy to prove that $\mathcal{B}(X\times Y) = ...
30
votes
2answers
547 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
0
votes
1answer
15 views

Can we deduce if a set is measurable, given a measurable function and a measurable space?

Let $f(x):X\rightarrow Y $, where $X$ is a measurable space. Suppose that $f$ is measurable. Let $E$ be a subset of $X$. Now, suppose that $f(E)$ is closed or clopen. Can we deduce that $E$ is a ...
2
votes
1answer
22 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
1
vote
0answers
21 views

Hölder continuity of measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function and $\mu$ the (via Stieltjes inversion formula) associated measure, which is a finite Borel measure on $\mathbb R$ and let $C(\lambda)$ be the function ($\alpha \in ...
0
votes
0answers
20 views

Conditionals of signed measures

My question pertains the definition of regular conditional measures of signed measures defined on product spaces. Consider a Suslin measurable space $\mathcal A=X\times Y$ with the Borel ...
0
votes
0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
1
vote
0answers
22 views

Fubini's theorem on a product of locally compact spaces which do not have countable bases

Let $X$ be a locally compact Hausdorff space. Let $\mathcal B$ be the $\sigma$-algebra generated by the family of open subsets of $X$. A measure $\mu$ on $\mathcal B$ is called a (positive) Radon ...
5
votes
2answers
193 views

Determining a measure through a class of measure preserving functions

Let $\mu$ and $\mu^\prime$ be probability measures over the sigma algebra $\Sigma$ consisting of the Lebesgue measurable subsets of $[0,1]$. Suppose also that $\mu$ and $\mu^\prime$ assign measure $0$ ...
4
votes
0answers
35 views

$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
0
votes
2answers
19 views

Expected Values of the product of a random variable and an indicator random variable

Let $X$ be a random variable $\in$ $L_{1}$ Given that $E[X]$ = $1$ , does that necessarily mean that : $E[X*1_{A}]$ = $P[A]$ ? My intuition is yes, since this is can be decomposed to $E[X]$ * ...
0
votes
0answers
17 views

Estimate for measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function (https://en.wikipedia.org/wiki/Nevanlinna_function) and let $\mu$ be the measure associated to $F$ via the Stieltjes inversion formula: ...
-3
votes
0answers
23 views

definition & properties of Lebesgue function [on hold]

I need the correct definition & properties of Lebesgue function to answer one question using Lebesgue function.
4
votes
2answers
1k views

Meaning of convergence in $L^1$ norm

What does it mean for a sequence $f_n$ to converge to some function, say, $f$ in the $L^1$ norm? Is it enough to show that $\int|f_n -f| \to 0$ or must one show as well that $f\in L^1$? I am ...
0
votes
1answer
36 views

Three questions about Haar measure

I have been reading on Haar measure recently. Let $G$ be a locally compact group with Haar measure $\mu$. $\mu(\{e\})>0$ then $G$ is discrete. $\mu(G)<\infty$ then $G$ is compact. we know ...
1
vote
1answer
17 views

Sub Sigma-Algebra and measurability

If a random variable $X$ is measurable with respect to a sub $\sigma$-algebra (let's say $\beta_{1}$), such that $\beta_{1}$ $\subset$ $\beta$ , is $X$ -necessarily- measurable with respect to the ...
0
votes
0answers
21 views

Continuous function such that range is different from the essential range.

Let $f$ be a function $\mathbb R \to \mathbb R$. The essential range EssRan(f) of $f$ is defined as the set of all numbers $z$ such that the preimage of every open ball around $z$ under $f$ has ...
2
votes
1answer
33 views

A piecewise $C^1$ curve has Jordan measure zero.

$\newcommand{\Reals}{\mathbb{R}}\gamma:[0,1]\to \Reals^2$ is an injective parametrization of a curve $\Gamma$, which is piecewise $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: ...