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

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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 ...
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1answer
10 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 ...
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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 ...
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1answer
10 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. ...
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1answer
26 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 ...
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2answers
24 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 $A$ ...
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2answers
45 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) = ...
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1answer
20 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 ...
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0answers
17 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 ...
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0answers
50 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 ...
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1answer
64 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 ...
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0answers
16 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 ...
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5answers
393 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 ...
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1answer
53 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 ...
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0answers
18 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 ...
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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 $H^2(\mu)$, ...
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1answer
23 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 ...
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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}: ...
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1answer
31 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 ...
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0answers
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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 $\mathcal{D}$ ...
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1answer
38 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 ...
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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 ...
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0answers
14 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 ...
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2answers
679 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
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1answer
27 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 ...
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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
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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 ...
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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 $x=y$ ...
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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 ...
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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 ...
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1answer
155 views

Borel measurable functions

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2 \to \mathbb{R}$ be defined by $h(x,y)=f(x)+f(y)$. Prove that $h$ is Borel measurable
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0answers
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Measurable functions on characteristic functions? [on hold]

Let $E\subset\mathbb{R}$ a measurable set. We define the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ where $f(x,y)=\mathcal{X}_{y+E}(x)$. Prove that $f$ is measurable in $\mathbb{R}^2$.
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1answer
23 views

Examples of functions where the Lebesgue integral as a measure is complete.

Let $f\in\mathcal{M}(\mathbb{R})$ non negative. For each $E\subset\mathbb{R}$ measurable we define $\mu_f(E)=\int_{E}f$. Prove (a) $\mu_f$ is a measure in $\mathcal{M}$ (b) Give an example of a ...
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2answers
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Stuck on existence proofs involving measurability and simple functions

Some classmates and I have been working through a sequence of problems in Royden's real analysis text, which are in the chapter on Lebesque measurable functions revolving around the Sequential ...
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stochastic variable constant on atoms = measurable?

Let $(\Omega, F, P)$ be a probability space, let $P_1,...,P_n$ be a partition of $\Omega$, let $F_0,...,F_n$ be a filtration of $F$, and let $P_t = \{ \text{atoms of } \ F_t\}$. I don't understand ...
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2answers
51 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
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0answers
30 views

Lebesgue dominated convergence theorem from RCA Rudin

Hello! This theorem from Rudin's RCA book. I would like to clarify some moments: $1)$ Since $\limsup\limits_{n} v_n\leqslant 0$ where $v_n=\int \limits_{X}|f_n-f|d\mu$. We get $0\leqslant ...
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1answer
25 views

A version of one-sided Chebyshev's inequality

Let $X$ be a real random variable with mean $\mu > 0$ and variance $\mu^2$. Does there exist a non-trivial upper bound on the probability $\Bbb P(X < 0)$ or is there a counterexample that shows ...
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1answer
34 views

Lebesgue integral, path connected and compact

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
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0answers
42 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 ...
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26 views

Reference for function being a distribution function

Let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$. Let $\nu$ be a probability measure on $(\mathbb R, \mathcal B(\mathbb R))$. Lastly let $F:\mathbb R \rightarrow [0,1]$ ...
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2answers
36 views

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to ...
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2answers
118 views

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
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What does it mean for two probability spaces to be distributed like each other? [on hold]

Given two probability spaces $A = (\Omega,F,P)$ and $B = (\Omega', F',P')$ what does it mean for "A to be distributed like B"?
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1answer
29 views

Indicator function integral

Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space. Let $A, B\in\mathcal A$. Assume that $\mathbb P(A) = 0.5$, $\mathbb P(B) = 0.4$ and $\mathbb P(A\cap B) = 0.1$. Find the integral over ...
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36 views

When a limsup can enter inside an integral

In the book "partial differential equation in classical mathematical physics" by I.Rubinstein,L.Rubinstein at page 411 I found something that I can't justify. It seems that the author (between ...
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Understanding the posterior of the Dirichlet process

Draws from a Dirichlet process (DP) are discrete, and exhibit clustering behaviour. Suppose I draw $G_{1:5}$ distributions from a DP. Then the posterior probability for $G_6$ is given by (Blackwell ...
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1answer
40 views

Please check whether the proof is correct or not.

Please check my solving. I want to know where to be wrong or illogical, or where logical jumps are. Problem Let $y=Tx$ be a nonsingular linear transformation of $\mathbb{R}^n$. If ...
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0answers
23 views

Questions on measurable functions and $L^p $ spaces

I'm learning about measure theory and $L^p$ spaces and need help with the following questions: $(1)$ True or False (justify): If $f : \mathbb R \to \mathbb{R}$ is measurable on $(-n, n), \, ...
3
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1answer
28 views

Lebesgue-$\sigma$-algebras $\mathfrak L^{p+q}\neq\mathfrak L^p \otimes\mathfrak L^q$

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...