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

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Caratheodory's Construction: Idea?

While reading Rudin's real and complex analysis I came across the following nice reasoning: Reasoning of Variation Measure Given a complex measure $\mu$ find its variation measure $|\mu|$ that is ...
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0answers
15 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
2
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1answer
14 views

Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Suppose $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are both complete measure spaces. Consider the following two measure spaces: $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ and $(X ...
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0answers
59 views
+50

Is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
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1answer
23 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...
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0answers
45 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
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2answers
40 views

Set of measure zero

Let $\mathcal{S}$ be the set $\mathcal{S} = \{(\mathbf{x}, \mathbf{y}) \in \mathbb{C}^{n} \times \mathbb{C}^{n} \mid \mathbf{x}^{H}\mathbf{y} = ||\mathbf{y}||^{2}_{2}\}$. Does $\mathcal{S}$ a set of ...
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0answers
17 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
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1answer
865 views

Proof of the Monotone Class Theorem

I am learning the Monotone Class Theorem from Jacod's Probability Essentials. I don't quite understand the idea of the proof in the book. I don't see the point in the proof at all. What's the use ...
2
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2answers
707 views

What's the difference between algebra and $\sigma$-algebra?

The title is quite misleading, I don't have a better one though. It's clear by definition that $\sigma$-algebra is also an algebra. Here is my question, for those algebras which are not ...
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6 views

spin-off of Choosing the correct subsequence of events s.t. sum of probabilities of events diverge [on hold]

Spin-off from here: Choosing the correct subsequence of events s.t. sum of probabilities of events diverge 1 Does m have to be 2? 2 Is it correct to say that for $(A_{nm+i})_{n\in\mathbb{N}}, m\in ...
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1answer
52 views

Using Borel-Cantelli Lemma

Let $X_1, X_2,\ldots$ be iid Geometric(p) where $p \in (0,1)$. Thus if $q=1-p$, then $P(X_n > k) = q^k$ for $k\geq 0$. Prove that for any fixed $\epsilon \in (0,1)$, CORRECTION: k is supposed to ...
0
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1answer
19 views

Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

Here is the problem. I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$. Is ...
2
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1answer
23 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
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0answers
10 views

Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
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0answers
26 views

$f(x)=\chi_{[0,1]}(x)$ a.e for a continuous function [on hold]

Prove that there NO exist a continuous function $f: \mathbb R \to \mathbb R$ such that $f(x)=\chi_{[0,1]}(x)$ a.e (under the lebesgue measure).
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3answers
96 views
+50

Probability of events in an infinite, independent coin-toss space

I am studying Steven E. Shreve's Stochastic Calculus book. Example 1.1.4 (p.4-6) constructs a probability measure on the space of infinely many coin tosses $\Omega_\infty$. In the example the ...
3
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3answers
58 views

Prove that every Lebesgue measurable function is equal almost everywhere to a Borel measurable function

Suppose $(\mathbb{R},\Sigma(m),m)$ is our measure space, where $m$ is Lebesgue measure. Also, suppose $f : \mathbb{R} \to [-\infty, \infty]$ is a Lebesgue measurable function. The problem: Prove ...
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1answer
54 views

Is this a $\sigma$-algebra(closed under contable union)?

Could I say that this $$ M=\{X\subseteq\Omega=[0,1):x\in X\iff y\in X\} $$ is an $\sigma$-algebra? I don't see whether it is closed under countable union. x,y are two singetons of $\Omega$ For ...
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0answers
32 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
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0answers
24 views

Billingsley “Probability and Measure” on constructing $\sigma$-fields

i'm starting to read, very slowly, Patrick Billingsley's "Probability and Measure". in chapter 1 "Probability", section 2 "Probability Measures", there's an optional section "Constructing ...
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1answer
34 views

Meaning of $P(Y|X=x)$

Suppose that $X$ and $Y$ are two random variables on $(\Omega, \mathcal H, P)$ with values in $(\mathbb R,\mathcal B_{\mathbb R})$. I want to understand what is "formally" the expression $P(Y|X=x)$ ...
1
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1answer
35 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
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1answer
77 views

Is true the boundary of compact set of $\mathbb{R}^n$ have Measure Zero?

Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \rightarrow [0, \infty[$ a measurable function. Suppose that there exist $C>0$ such that $$\int_K f dm < C,\ \forall\ K\subset\Omega,\ K\ ...
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1answer
23 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
0
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1answer
32 views

Every Lebesgue measurable set contains a closed subset such that the set difference has small measure

M is the lebesgue measurable sets on $\mathbb{R}$. I have this exercise: Suppose that $E \in M$. Show that for each $\epsilon > 0$, there is a closed set F, with $F \subset E$ and $\lambda(E ...
4
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1answer
40 views

$\int f = \lim\int f$ but $\int_{E}f\neq\lim\int_{E} f_{n}$

This is exercise 2.13 in Folland's Real Analysis textbook Let $(X, \mathcal{M})$ be a measurable space. Suppose $\{f_{n}\}\subset L^{+}$, $f_{n}\to f$ pointwise, and $\int f=\lim\int ...
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0answers
34 views

Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains an open interval [duplicate]

Suppose that $E$ and $F$ are Lebesgue measurable sets of $\mathbb{R}$, and their Lebesgue measures $m(E) > 0, m(F) > 0.$ Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains a nonempty ...
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1answer
48 views

Sum of random variable

Considering two continuous random variables $X$ and $Y$ with $d.f \; F_X, F_Y$ I want to fin the distribution and distribution function of the sum $Z=X+Y$. \begin{align} P\{Z \leq z\} &= P\{X+Y ...
2
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1answer
19 views

Haar measure on Upper triangular unipotent matrices in $GL_n(\mathbb{F})$

I am reading Bump's book on Automorphic forms and Representations. I don't have a clear understanding of Haar measures and so, I am finding it difficult to do some of the exercises. Can somebody help ...
4
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1answer
208 views

Proof to motivate the need for measure theoretic probability

Im using Billingsley's Probability and Measure. In lecture the instructor motivated the need for measure theory in probability by providing a solution to the following problem: Show that a ...
2
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1answer
40 views

If $\varphi f\in L^1(\mu)$ for every $f\in L^1(\mu),$ then $\varphi \in L^\infty$

Let $\varphi$ be a measurable function for which $\varphi f\in L^1(\mu)$ for every $f\in L^1(\mu).$ Show that $\varphi \in L^\infty(\mu).$
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0answers
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Prove $\mu(\{x:f(x)>t\})=m(\{s>0:f^*(s)>t\})$ for every $t>0.$ [on hold]

Let $f$ be positive measurable function on space $X$ with $\sigma$ finite measure $\mu$ for which $\mu (\{x:f(x)>t\})<+\infty$ for every $t>0$. Define $f^* (s)=sup\{s\geq ...
0
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1answer
27 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
0
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1answer
13 views

What is meaning of symbol $\wedge$ in Probability with Martingales by Williams

On page 62 of probability with Martingales by Williams, he defines: For $n \in \mathbb{N}$, define $X_n(\omega) := \{ |Y(\omega)| \wedge n\}^p$ I know $\wedge$ in the context of set theory, ...
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2answers
28 views

Measures: Sigma-Additivity vs. Continuity

Let $R$ be a ring of sets that contains the empty set and $\mu$ be a positive and finite set function on $R$. If $\mu$ is countable additive, then it is continuous from below and above: $$A_n\uparrow ...
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1answer
32 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
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2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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0answers
9 views

Exterior measure of a subset $A \subset \mathbb R_n$ equals the measure of a$G_{\delta}$

Let $A \subset \mathbb R^n$, prove that there is $H$: $A \subset H$, with $H$ a $G_{\delta}$ set such that $|A|_e=|H|$. The definition of $|A|_e$ is $|A|_e=\inf\{m(U): A \subset U\}$ where the ...
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2answers
32 views

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
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2answers
40 views

$L^{\infty} (X, \mu)$ is not separable? [on hold]

Show that $L^{\infty} (X,\mu)$ is not separable if $X$ contains sequences of disjoint sets of strictly positive measure?
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votes
3answers
565 views

Doubt in Scheffe's Lemma

While reading up on "Glivenko Cantelli Theorem" from Probability Models by K.B Athreya, the author used 2 lemmae to prove it. One was called Scheffe's lemma, the other Polya's theorem. Scheffe's ...
2
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1answer
45 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
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1answer
20 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
6
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1answer
59 views

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me ...
2
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1answer
17 views

Does simply-connected imply measureable?

The famous examples of non-connected sets involve a sophisticated selections of points from a ball (or another object). This raises the following question: if a certain object in a Euclidean space is ...
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1answer
30 views

open set $O$ such that $\partial(\overline{O})$ has positive measure

Find an open set $O$ such that $\partial(\overline{O})$ has positive measure. The hint is to consider a Cantor set, with positive measure. But that does not work, because all the Cantors are closed ...
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1answer
44 views

Halmos Measure Theory section 39 Theorem D

I have trouble explaining the remark "The function $\phi$ plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals". I know ...
4
votes
2answers
953 views

Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
2
votes
1answer
384 views

Show that $L^1$ convergence implies uniform integrability.

Let $(X,\mathscr{M}, \mu)$ be a measure space and suppose $(f_n)$ is a sequence of $L^1$ functions ($L^1$ being the space of equivalence classes of absolutely integrable functions) converging to $f$ ...