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

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155 views

Is $\nu_1\perp\nu_2$ equivalent to $|\nu_1|\perp|\nu_2|$ for complex measure?

Suppose $\nu_1$ and $\nu_2$ are two complex measures, do we have $\nu_1\perp\nu_2\Leftrightarrow|\nu_1|\perp|\nu_2|$? if yes, please give a proof. If no, please give a counterexample. Thanks! PS: two ...
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438 views

Different definitions for Lebesgue points

Let $f \in L^1_{loc}$. We call $x \in \mathbb R^n$ an Lebesgue point, if $\lim \limits_{R \rightarrow 0} \frac{1}{m(B_R(x))}\int_{B_R(x)} f \;\;\;\;$ exists or $\lim \limits_{R \rightarrow 0} ...
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157 views

Measure theoretic proof on composition of invariant functions and integrability

I'm looking for a proof that if f is $\mu$-integrable ($\mu(|f|) < \infty$, where $\mu(f)$=sup{$\mu(g):g \leq f,\ g \text{ simple}$}), and $\tau$ is measure preserving ($\tau^{-1}(A)$ measurable ...
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185 views

Sigma finite and construction of map

Let $X$ be a sigma finite measure space with $p>0$. The book states the following: There exists a map $f\in L^{p}(X)$ such that $f>0$ and $f$ is bounded by $1$. Proof: Let $C_{n}$ be a ...
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113 views

Quotients , L^p

As usual denote $L^p$ the quotient space where two integrable functions are identified if they are equal almost everywhere. So I'm using the definition written here: ...
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1answer
1k views

limit superior and limit inferior of the given sequence of sets

A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
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20 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated random variables $\Omega\to [0,\infty]$ with ...
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1answer
37 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
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17 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
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14 views

Probability measure of rank-$r$ matrices

I have a question about the distribution of matrices with a specific rank. Consider $\mathcal{M}^{n\times m}$ the set of all $n \times m$ matrices with entries in some field $\mathbb{K}$. If I define ...
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36 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
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53 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
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35 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
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43 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
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27 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
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33 views

Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
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15 views

Requirements for existence Lebesgue-Stieltjes measure corresponding to distribution function in $\mathbb{R}^n$

I am going through Ash's book "Probability and Measure Theory". It says that: We know that a distribution function of $\mathbb{R}$ determines a corresponding Lebesgue-Stieltjes measure. This is true ...
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3answers
22 views

Indicator in expectation

Suppose we have a measure space $(\Omega,\mathcal{F},P)$, say we have a random variable $X$ defined on this measure space. My question now is; if we have an event say $F \in \mathcal{F}$ is it in ...
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17 views

Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in ...
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20 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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24 views

measurable set where limit of functions exists

Let $f_n :X\to [-\infty,+\infty]$ be measurable functions. Is $ E = \{ x\ \vert \lim\limits_{n\to\infty} f_n (x) \text{ exists}\} $ measurable?
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33 views

Lebesgue integrable function in rationals

Function $f : [0,1] \to \mathbb{R}$ defined as $ f(x) = \begin{cases} 1 & x\notin\mathbb{Q}\\ 0 & x\in\mathbb{Q} \end{cases} . $ As is well known $f$ is not integrable in the Riemann sense. ...
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1answer
23 views

Understanding a proof using Caratheodory's criterion

I'm reading a proof on proving that countable unions of measurable sets are again measurable. The proof is as follows Let $T$ be a test set. $E_1\cup E_2$ is measurable if Caratheodory's ...
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32 views

Amalgamated product of two measures.

Please do not get annoyed by the symbols below. The problem has a really simple statement. $2^X$ denotes the set of functions from $X$ to $\{0, 1\}$ equipped with usual product topology. By $Fn(X, ...
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40 views

What is a convolution kernel?

What is a convolution kernel? (in measure theory, probability theory) In which book can I read about kernels on measurable spaces and convolution kernels? Thank you!
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21 views

Continuity of set function on field and relation with continuity in topological space

I am trying to understand how continuity of measures relates to the definition of continuity in topological sets : Every open set in range corresponds to an open set in domain. A real valued set ...
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79 views

A problem with kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ be two measurable spaces. A $kernel$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is an application $N : p \mathcal{B} (E) \rightarrow p ...
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37 views

Show $\int_{\mathbb R} f * g \ d \lambda = \int_{\mathbb R} f \ d \lambda \cdot \int_{\mathbb R} g \ d \lambda$.

Suppose $f,g \in \mathcal L_{\mathbb C}^1(\lambda)$. Show $\int_{\mathbb R} f * g \ d \lambda = \int_{\mathbb R} f \ d \lambda \cdot \int_{\mathbb R} g \ d \lambda$. I see that $\int_{\mathbb R} ...
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1answer
89 views

Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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13 views

Coarea formula for fractional dimension

The coarea formula states that any locally Lipschitz function (e.g. a $C^1$-function) $F:\mathbb{R}^N\to\mathbb{R}^n$ with $N\geq n$ satisfies $$\int_A JF(x) \mathrm{d}\mathcal{H}^N = ...
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24 views

Prove that if $f$ is monotone, then the four Dini derivatives of $F$ are measureable

The Dini derivatives of $F$ at $x$: (i). The upper right derivative $\overline{D^+}F(x):=\limsup\limits_{h\to ...
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38 views

Is this definition of Lebesgue integral problematic?

Folland defines the integral of a non-negative measurable function $f$ to be $$\int f d\mu = \sup \left\{ \int \phi d\mu : 0 \leq \phi \leq f, \phi \text{ simple}\right\}$$ after defining the ...
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29 views

Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, ...
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1answer
26 views

an inequality on $L_p$ and $l_2$

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that ...
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33 views

Monotone convergence theorem allows the limit to be infinity

Monotone convergence theorem(MCT) doesn't impose any restriction on the limit. For example, if $\{X_n\}$ satisfies $0 \le X_n \nearrow X$ with $EX=\infty$, then I still could use MCT to get ...
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14 views

Definition of Measure regular

Book's, Real and complex analysis, Walter Rudin. I am somewhat confused. My question is: "In other words, we are looking at $L^p$, where $\mu$ is Lebesgue measure on $[0,2\pi]$(or on $T$), ...
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2answers
24 views

Measure of the intersection of a ball and a compact subset

Do you know a large class of compact subset $K$ of $\mathbb{R}^d$ such that for each such compact $K$, there exists a $r>0$ with $\inf_{x \in K} \lambda^d(B_r(x) \cap K) > 0$, where $\lambda^d$ ...
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36 views

How is this passage probably meant?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}^d}$. Let $\mathfrak{B}$ denote the Borel field on $X$ generated by its topology and let $\mu_{p_0,p_1,p_2}$ be product measure on $X$ in which the $i$'s have ...
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What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability?

What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability? I know Borel measurability implies both Lebesgue-Stieltjes measurability and Lebesgue measurability. ...
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16 views

Haar measure of an angle-distance ball in SO3

If for rotations $R_0$, $R_1$ we define the distance $d(R_0, R_1)$ to be the angle of $R_0 R_1^{-1}$ and given $r\in [0,\pi)$, what is the "volume" (normalised Haar measure) in $SO_3$ of the ball ...
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24 views

Independent events in “coin tossing space”

This question really puzzles me. The setup: We define the "coin tossing space" by starting with the set $\Omega = \{ 0,1 \}^{\mathbb N}$ and then defining the finitely determined events $t \in ...
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20 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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28 views

Why is $g(X)$ measurable with respect to $\sigma(X)$

Suppose $X,Y$ independent random variables and $\phi$ be a function such that $E[\phi(X,Y)]<\infty$. Let $g(x)=E[\phi(x,Y)]$. We need to show that $g(X)=E[\phi(X,Y)|X]$. However I cant show ...
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57 views

Question on $L_p$ spaces

Consider $L_p = L_p(\lambda^n)$ with the Lebesque measure on $\mathbb{R}^n$ and $1 \leq p < \infty$. Let $f_0(x) = |x|^{-\alpha}$ if $|x| < 1, f_{0}(x) = 0$ for $|x| \geq 1$. Show that: $f_{0} ...
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18 views

Understanding a proof of Lusin. Question about measure of the set discontinuity point of a step function defined on finite measurable sets.

I'm reading a proof and I don't understand the red part. I think that if $f_n$ is a step function, then the set of points in which $f_n$ is discontinuous is countable, in particular has $0$ lebesgue ...
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29 views

Commutativity of Lebesgue-Stieltjes convolution

Let $F,G$ be non-decreasing real functions of bounded variation on $\mathbb{R}$ and $\mu_F$ the Lebesgue-Stieltjes measure defined by it. Kolmogorov-Fomin's says (p. 452 here) that we can commute the ...
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34 views

Can someone check my answer to a measure theory question on existence and equality of three integrals.

I have been told to investigate the existence and equality of the integrals; $\int_{[0,1]^2} f\;d\lambda^2$, $\int_0^1\int_0^1 f\;d\lambda(x)d\lambda(y)$ and $\int_0^1\int_0^1 ...
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37 views

Determining measurable sets

STATEMENT: Let $α$ be the non-decreasing function on $\mathbb R$ defined by $α(t) = 0$ if $t ≤ 0$ and $α(t) = 1$ if $t > 0$. Let $μ_α([a,b))=\alpha(b)-\alpha(a)$ , with $μ^*_α$ the corresponding ...
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60 views

A basic question about the convergence a sequence of measurable functions

Let $(X,\mathcal{B},\mu)$ be probability space i.e. $\mu(X)=1$. If $\{g_n\}_{n=1}^{\infty}$ is a sequence of measurable functions such that $\Sigma_n \int g_n^2 d\mu <\infty$ then $g_n \to 0$ ...
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37 views

Probability that a set is a subset of another set and limits of probabilities

Let $\{A_n\}, \{B_n\}$ be two sequences of infinitely many finite sets $A_n, B_n$ satisfying $$ \lim_{n \to \infty} \mathbb{P}(a \in B_n \ \mid \ a \in A_n) = 1, $$ where, for each $n$, the element ...