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

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3
votes
1answer
413 views

Independence $\sigma$-algebras

Id like to prove that $\sigma(X_{1},...,X_{n})$ and $\sigma(X_{n+1},...)$ are independent for independent random variables $X_{i}$. I've found that $\{X_{1}\in B_{1},...,X_{n}\in B_{n}\}$ generates ...
0
votes
1answer
34 views

Lebesgue measure as $\sup$ of measures of contained compact sets

I know, from Kolmogorov-Fomin's Элементы теории функций и функционального анализа, the definition of external measure of a bounded set $A\subset \mathbb{R}^n$ as $$\mu^{\ast}(A):=\inf_{A\subset ...
0
votes
0answers
2 views

amalgamation of three measures

Let $X, Y, Z$ be three measurable (say, Radon) spaces, and let $\mu_{ab}, a, b \in \{X,Y,Z\}, a \neq b$ be three measures on spaces defined on respective spaces $a \times b$. Is it true that there ...
2
votes
2answers
50 views

The Dirac delta does not belong in L2

I need to prove that Dirac's delta does not belong in $L^2(\mathbb{R})$. First, I found the next definition of Dirac's delta $\delta :D(\mathbb R)\to \mathbb R$ is defined by: ...
2
votes
0answers
14 views

Esscher Transform extended

The Esscher-transform is a well know tool in the financial section. I posted this in statistics also, since it relates to continuous sampling. Im not sure if my approach is right, so it would be nice, ...
2
votes
1answer
33 views

Problem on $\sigma$-algebra from Rudin

Does there exist an infinite $\sigma$-algebra which has only countably many members ? Proof: Suppose that $\sigma$-algebra $\mathfrak{M}$ has countably many members, namely $\{A_i\}_{i=1}^{\infty}$. ...
1
vote
1answer
20 views

Iterated integral and integrability

Hi everyone: Suppose $B_{1}$ and $B_{2}$ are balls in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ (let's say for $m,n\geq2)$. Suppose that $f(x,y)$ is defined and measurable eveywhere. Beside, $$0\leq ...
-2
votes
0answers
14 views

Doubts on a proof of Hanh's decomposition theorem

I am having a bad time trying to figure out why does the inequality following "It follows that every measurable subset F..." holds. I'm sorry if it's a dumb question. ...
8
votes
0answers
113 views

Fast convergence in $L^1$ implies convergence almost everywhere

This is a proof-verification request. Claim: Let $(X,\mathscr M,\mu)$ be a measure space. Let $f_n$ ($n\in\mathbb N$) and $f$ be measurable, integrable, real-valued functions such that ...
0
votes
1answer
24 views

Two Borel disjoint sets such that the perimeter of union is less than the sum of perimeters

Exercise: Find two Borel disjoint and bounded sets $E, F \subset \mathbb R^n$ such that $\operatorname{Per}(E) + \operatorname{Per}(F) > \operatorname{Per}(E \cup F)$. ($\operatorname{Per}(A)$ is ...
0
votes
0answers
30 views

Perimeter of the Unit Ball in $\mathbb R^n$

Exercise: Calculate the perimeter of the unit ball in $\mathbb R^n$, i.e. show that $\mathcal H^{n-1}(S^{n-1})=n\omega_n$, where $\mathcal H^{n-1}$ is the Hausdorff measure of dimension $n-1$ and ...
1
vote
2answers
47 views

Theorem 1.40 from Rudin RCA

Hello! I read this theorem and understood it. But let me ask you one question: where did he use that $\mu(X)<\infty$? Is this condition crucial?
11
votes
1answer
291 views
+100

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
-1
votes
1answer
32 views

Proof of the existence of $E(X|\mathcal{G})$

I am looking through my lecture notes, which follows Billingsley, regarding the proof of the existence of $E(X|\mathcal{G})$. The theorem is: Let $(\Omega, \mathcal{F}, P)$ be a probability space, ...
-1
votes
0answers
25 views

Why is $P(A|\mathcal{G})$ $\mathcal{G}$-measurable? [on hold]

I want to show that $E(1_{A}|\mathcal{G})=P(A|\mathcal{G})$, for every $A \in \mathcal{F}$, where $1_{A}$ is the indicator function and $\mathcal{G}$ is a $\sigma$-subfield of the $\sigma$-field ...
0
votes
2answers
35 views

External measure invariant under unitary transformations

Let us define the external measure of the set $A\subset \mathbb{R}^n$ as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of $A$ ...
3
votes
0answers
36 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 $g\in L^1(\mu)$. ...
0
votes
0answers
12 views

Two definitions of Lebesgue measurability

Let $E$ be bounded interval of $\mathbb{R}$. 1st definition: Subset $A\subset E$ is measurable if and only if $m^{*} (A)+m^{*}(E\setminus A)=m^* (E)$ 2nd definition: Subset $A\subset E$ is ...
3
votes
0answers
22 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 ...
1
vote
1answer
31 views

Property of function $f=0$ almost everywhere

Theorem: Suppose that $f:X\to [0,\infty]$ is measurable, $E\in \mathfrak{M}$, and $\int \limits_{E}fd\mu=0$. Then $f=0$ a.e. on $E$. It's very famous fact which can be proven easy. Am I right that ...
1
vote
2answers
51 views

Find a subset of A such that its boundary does not have measure zero

Question Find a subset $A$ of $[0,1]$ such that $A=cl(intA)$ and yet $bd(A)$ does not have measure $0$. I don't know how to construct it. I think it should be closed set, cannot be empty by ...
2
votes
0answers
22 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 ...
1
vote
1answer
33 views

Real Analysis, Folland Proposition 1.7 elementary family

Definition - An elementary family is a collection $\varepsilon$ of subsets of $X$ such that i.) $\emptyset\in \varepsilon$ ii.) if $E,F\in \varepsilon$ then $E\cap F\in \varepsilon$ iii.) if $E\in ...
2
votes
2answers
227 views

Saturated measure defined as a supremum of a semifinite measure and countable unions

Here is what I am working on: Suppose that $\mu$ is semifinite. For E in $\overline{M}$, define $\underline{\mu}(E)=\sup\{\mu(A):A$ in $M$ and $A \subseteq E$$\}$. Then $\underline{\mu}$ is a ...
1
vote
1answer
11 views

Compositions preserving measurability

My question is based on this post, which I summarize below. Claim: Let $(X, \textbf{X})$ be a measurable space, $f:X \to \mathbb{R}$ is X-measurable, $g: \mathbb{R} \to \mathbb{R}$ is continuous. ...
3
votes
1answer
39 views

Corollary of Lebesgue's DCT from Rudin

Hello! After attentive reading of this theorem I have some questions: $1)$ If $E=\{x\in S: \varphi(x)<\infty\}$ then what is $E^c$? I know that it's the complement of set $E$, i.e. $E^c=\{x\in ...
1
vote
2answers
48 views

Real Analysis, Folland Theorem 1.9, extention of a measure to a complete measure

I have posted this theorem before but I am re-posting it again because I have a different question. Theorem 1.9 - Suppose that $(X,M,\mu)$ is a measure space. Let $\mathcal{N} = \{N\in M:\mu(N) = ...
0
votes
1answer
22 views

Measurable sets on Polish spaces

A measure on a Polish space $X$ will be refered as a function $\mu:BOREL(X)\rightarrow [0,1]$ s.t. $\mu(\emptyset)=0,\mu(X)=1$ If $\{A_n:n\in\omega\}\subseteq BOREL(X)$ is a sequence of pairwise ...
1
vote
0answers
19 views

Series of integrable functions converges pointwise almost everywhere

I need some help, solving the following problem I found in my textbook. QUESTIONS APPEAR IN BOLD CAPITALS. Let $(X,\Sigma,\mu)$ be a measure space and $f_n \colon X \to \mathbb{C}$ ($n \in ...
0
votes
0answers
52 views

Small filters are measurable

i want to show, that a filter $\mathcal{F}$ on $\omega$ (considered as a subset of $2^\omega$), which is small, is measurable. I found a lemma (without proof), that every small set is null. So, if ...
0
votes
1answer
65 views

what does $\frac{\text{d}x}{x}$ mean?

I saw in a lecture recently the Gamma-function written like $$\Gamma (k) = \int_0^\infty e^{-x} x^k \frac{\text{d}x}{x}$$ and the professor said, that the integral was with respect to the measure ...
0
votes
0answers
17 views

Extension of definition measurable function

This is from Rudin's RCA book. I read this paragraph few times with big aatention and have some questions: $1)$ Why Rudin's enlarges the definition in above manner? $2)$ He wrote that domain of ...
25
votes
5answers
400 views

What is wrong in this proof: That $\mathbb{R}$ has measure zero

Consider $\mathbb{Q}$ which is countable, we may enumerate $\mathbb{Q}=\{q_1, q_2, \dots\}$. For each rational number $q_k$, cover it by an open interval $I_k$ centered at $q_k$ with radius ...
5
votes
1answer
54 views

Definition of measurable space - sigma algebra

A measurable space is a set $S$, together with a nonempty collection, $\mathcal{S}$, of subsets of $S$ satisfying the following two conditions: For any $A$, $B$ in the collection of ...
1
vote
0answers
28 views

Nonanticipativity constraint (filtration/measure theory)

I am trying to show that stochastic process must attend the nonanticipativity constraint using filtration in measure theory. Adaptability of a stochastic process tell us that: $$\sigma(X_t)\subset ...
0
votes
0answers
11 views

Subnormal Weighted shift and First order derivative

Let $\mathbb B^m$ denote the Eucledian ball in $\mathbb C^m.$ Does there exist a reinhardt measure $\mu$ supported on $\partial \mathbb B^m,$ the boundary of ball, so that the Hilber space $H^2(\mu)$, ...
1
vote
1answer
27 views

Approximating Lipschitz Functions by $C^1$ functions

According to Evans-Gariepy as a corollary of the Whitney's Extension Theorem we have the following Theorem (Approximating Lipschitz Functions) Suppose $f: \mathbb R^n \to \mathbb R$ is Lipschitz ...
5
votes
1answer
56 views

Converse for Fubini-Tonelli's theorem

By Fubini-Tonelli's theorem, we know that if $E\in \mathbb{R^{n+m}}$ and $f: \mathbb{R^{n+m}}\to \mathbb{R_{>0}}$ are measurable and $f$ integrable, then the sections $E_x=\{y\in \mathbb{R^m}: ...
3
votes
1answer
32 views

A characterization of Borel measurability

I need help proving the following fact. Let $(X,\textbf{X})$ be a measurable space. Then $f:X \to \mathbb{R}$ is X-measurable iff $f^{-1}(E) \in \textbf{X}$, $\forall E \in \textbf{B}$. Defns and ...
0
votes
0answers
16 views

Measurability of translated sets on a space of cadlag functions

Let $(D[0,T], \mathcal{D})$ be the measurable space of real-valued functions on the interval $[0,T]$ which are right continuous and have left limits, equiped with the $\sigma$-algebra $\mathcal{D}$ ...
0
votes
1answer
39 views

Measure on function space

Suppose $F$ is a collection of continuous functions on $[0,1]$ (with the $\sup$ metric) and $\mu$ is a probability measure on $F$. Is it true that the mapping $x \mapsto \int_F f(x) \, d\mu(f)$ is ...
2
votes
0answers
31 views
+50

algebraic sum of a graph of continuous function and itself - measure > 0 imply nonempty interior?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Then $G+G$ is compact: algebraic sum of a graph of continuous function and itself Borel or ...
0
votes
0answers
15 views

function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
2
votes
2answers
681 views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and ...
3
votes
1answer
29 views

Can Haar Measures Exist On Not Locally Compact Spaces?

In a reading course on measure theory this semester I had the pleasure of preparing a lecture covering the existence-uniqueness of Haar measure on locally compact groups. Since the proofs (as ...
8
votes
2answers
1k views

Folland, Chapter 1 Problem 17

Problem 17: If $\mu^*$ is an outer measure on $X$ and $\{A_i\}_{i=1}^{\infty}$ is a sequence of disjoint $\mu^*$-measurable sets, then $\mu^*(E\cap \cup_{j=1}^{\infty} A_j)=\sum_{j=1}^{\infty}(E\cap ...
4
votes
1answer
261 views

Please help in Folland Analysis Proposition 2.11

I don't really understand proposition 2.11 in Folland. So please help me to explain, as well as give some hints to prove it. The part makes me confuse is $f=g$ $\mu-a.e$ ,does it means f equals to g ...
1
vote
1answer
34 views

$\mu$-completion of $\sigma$-algebra

Rudin shows that $\mu$ is well-defined on $\mathfrak{M}^*$. But this little bit confuses me since well-defined of function I understand in the following meaning: $f:X\to Y$ is well-defined if $x=y$ ...
1
vote
2answers
60 views

Question about proof extending measure to complete measure

I am looking through a proof in Folland, for Theorem 1.9, which states: Suppose that $(X, M, \mu)$ is a measure space. Let $N = \{N' \in M : \mu(N') = 0\}$ and $M' = \{E \cup F : E \in M' \text{ and ...
2
votes
1answer
69 views

Uniformly integrable implies integrable?

The term "uniformly integrable" sounds (to a layman like me) to be stronger than integrable. Just like how uniformly convergent is stronger than simply being convergent. However, from the definition ...