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

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1answer
12 views

Is it sufficient to compare two measures on a generator?

Let $\mu_1$ and $\mu_2$ be two measures defined on a common sigma algebra $\Omega$, and let $\mathcal{G}$ be a non-trivial generator of $\Omega$. If $\mu_1(A) \leq \mu_2(A)$ for every element $A \in ...
0
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0answers
11 views

Volterra operator is completely continuous

Let $\mu$ be the Lebesgue measure on $[0,1]$ on the borelians, and consider the Volterra operator $V:L^1[0,1]\to C[0,1]$ given by $$ Vf(t)=\int_0^t f d \mu $$ So, I want to show the following ...
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0answers
16 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
1
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2answers
33 views

Positive measure and $L^2$ space

I have a question about measure theory. Let $(E,\mathcal{B})$ be an arbitary measurable spase and let $m$ be a positive measure on $(E,\mathcal{B})$. Let $f \in L^{2}(E;m)$ (fix). $0 \leq\int ...
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0answers
31 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
27 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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1answer
37 views

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
49 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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1answer
25 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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2answers
42 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
18 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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0answers
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
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 ...
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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).
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
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 ...
1
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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
36 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 ...
3
votes
3answers
60 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 ...
1
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1answer
78 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
24 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 ...
2
votes
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 ...
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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 ...
1
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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 ...
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0answers
15 views

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 ...
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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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 ...
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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, ...
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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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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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
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
63 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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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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3answers
101 views

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 ...
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 ...
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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
41 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?
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
18 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
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0answers
27 views

Borel image of the projection [closed]

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps Borel sets to Borel sets?
5
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1answer
131 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
2
votes
1answer
36 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
0
votes
2answers
26 views

A proposition about positive random variables and expected values

I have problems to give a proof for the following proposition: Consider a random variable $X$ with values in $[0,+\infty]$. If $P(X=+\infty)>0$, then $E(X)=+\infty$ (notation: $E(X)=\int X ...
4
votes
2answers
221 views

Is this set measurable?

Let $E$ be a subset of $\mathbb{R}$. Assume that $\forall x\in E, x$ is a limit point of $E\setminus\{x\}$. Then, is $E$ Lebesgue-measurable? For example, any perfect subset, open subset or ...