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

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Understanding the Definition of Lebesgue space (as it is given in a book of Katok/ Hasselblatt)

On page 733 of Modern Theory of Dynamical Systems by Anatole Katok and Boris Hasselblatt I found the following definition of Lebesgue space: Definition A.6.4. A measure space ...
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39 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
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119 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
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82 views

When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten-von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, ...
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72 views

Existence of a nonmeasurable set with slices of measure zero

Does there exist a nonmeasurable set $A\subseteq \mathbb{R}^2$ such that for each line $l$ in the plane the intersection $A\cap l$ has one dimensional measure zero. (What about the problem for only ...
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1answer
83 views

Sigma algebra on space of signed Radon measures

consider the space $M = \left\{ \mu : \mathscr{B}(\mathbb{R}) \to \mathbb{R} \cup \left\{ -\infty, +\infty \right\} \ | \ \mu \text{ signed Radon measure} \right\}$ which is not a vector space, since ...
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136 views

The projective limit of probability spaces and the Kolmogorov-Daniell theorem

Does the "projective limit" concept exist for probability spaces? The only result that I know of seems to be the Kolmogorov-Daniell theorem, but this is just a particular case where the spaces ...
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79 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
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1answer
31 views

Gauge Integral: Non-Borel Spaces

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
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52 views

Asymmetry in definition of regular measure

In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if \begin{equation} \mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\} \end{equation} and ...
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1answer
478 views

Sequence of measurable functions converging a.e. to a measurable function?

I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e. ...
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1answer
35 views

Problem involving decomposition of measures

Let $\mu$ be a signed measure. We wish to prove that $$\left| \int{f} \> d\mu \right| \leq \int{|f|} \> d|\mu|.$$ (We are given the following defintion: $\int{f} \> d\mu = \int{f} \> ...
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24 views

Pushfoward of measures Lipschitz continuous in total variation

Let $X,Y$ and $Z$ be metric spaces and $f:X\times Y\to Z$ be a measurable map. Suppose that we are given a probability measure $\mu$ on $Y$, and define a stochastic kernel $$ ...
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46 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
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67 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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100 views

Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
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1answer
77 views

Poincaré Recurrence Theorem (measure theory version)

I had a look on the proof of the following Recurrence Theorem of Poincaré: Let $(\Omega,\Sigma,T,m)$ be a conservative dynamical system in measure theory for which the function $T^{-1}$ ...
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1answer
86 views

Does Vitali set imply the axiom of choice

I know that the construction of Vitali set needs the axiom of choice, but this only states that $AC \implies V$. Is it also true that $V \implies AC$? If $\neg AC \implies \neg V$, then what ...
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69 views

A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu $ and $\lambda $ are measures in $M$ st $\mu \ge $ $\lambda $ then show that $m$ defined as $\mu= \lambda + m $ is a measure 2) Prove that if ...
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1answer
85 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
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1answer
60 views

Measure Theory - working with unusual measures and set functions

Let $m$ define the Lebesgue measure. Let $\mu$ define the measure $\mu(A)=m(A\cap(0,1))$ for a Borel set $A$. Let $K=\bigcap \{A:A$ is closed, $\mu(A)=1\}$, $D=\bigcap \{G:G$ is open, ...
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66 views

Approximation of integration by simple functions.

Let $f: \Omega\longrightarrow \mathbb{R}$ be a Lebesgue integrable function. Does $$ s_n=\sum_{-\infty}^\infty\frac{k}{2^n}\lambda\left\{\frac{k}{2^n}<f\leq \frac{k+1}{2^n}\right\} $$ ...
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46 views

What is exactly meant by a countable collection in the defintion of a sigma-algebra

This a just a small question about the definition of a sigma-algebra and what is eaxactly meant by countable? Would be grateful for any clarification on this. Most texts define a sigma-algebra ...
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107 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
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1answer
61 views

Why this function is continuous?

Let $(\Omega,\Sigma,\mu)$ be a sample space and let $L^2= \lbrace f:\Omega \rightarrow R / \int f^2d\mu <\infty \rbrace$ be a Hilbert space. Let $L_n=L^2\times L^2 \times .... \times L^2$ ($n$ ...
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1answer
45 views

strengthen the condition of convergence in measure of sequence of functions

Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative measurable functions on a measurable set $E$. (1). Suppose for any $\epsilon>0$, $$\sum_{n=1}^\infty \mu\{x\in E: ...
2
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1answer
27 views

Convergence in distribution problem

I want to prove that, in $(\mathbb{R},B(\mathbb{R}))$, we have that $\frac{1}{n}\sum_{i=1}^{n}\delta_{\frac{i}{n}}$ converges to $U_{[0,1]}$. We need to prove, by definition, that $\lim_{n \to ...
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1answer
60 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
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62 views

Noob Question : Need help to understand : Probability with Martingales : page 24

Coin toss infinitely often. The sample space is $\Omega = \{ H , T \}^{\mathbb{N}} $ And we do not have any problem. And a typical point is $\omega = \omega_1 \omega_2 \omega_3 ... $ where $\omega_n ...
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104 views

Trace of a measure on a subset (or restriction of a measure to a subset)

This is an exercise from Measure Theory by Cohn. Given a measurable space $(X,\mathcal{A})$ and subset $C$ which may not be measurable, we can form the trace of $\mathcal{A}$ on $C$ denoted ...
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76 views

Equality of measures, symmetric intervals, compact, bounded, measurable sets

Let $\mu$ be a measure on $L_m$ , ($m \ge 1$) - $\sigma$ - algebra of Lebesgue-measurable sets, such that its values on compact symmetric intervals (cubes) are equal to Lebesgue measure of those ...
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1answer
130 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
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107 views

Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
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69 views

Distribution function of a Compound Poisson process

Is it possible to find a sort of closed form distribution function of this random variable: $$ Z= \sum_{k=1}^{N}X_kB_kC $$ where $N \perp (X_k)_{k\in \mathbb{N}} \perp (B_k)_{k\in \mathbb{N}} \perp C ...
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98 views

A measure has no point masses: is it absolutely continuous?

I have a question about measure theory. Let $\mu$ be a measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R})$. Assume that $\mu$ has no point masses - i.e. for every $a \in \mathbb{R}$, $\mu({a})=0$. Can ...
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49 views

Does convergence of integrals imply a.e. convergence of functions?

Let $\{u_n\}_{n\in\mathbb{N}}\subseteq L^{\infty}([0,T],U)$ be a sequence of measurable functions, where $U=[0,\overline u]$ is a compact set, and suppose that $\lambda(t,du)$ is a probability kernel ...
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1answer
107 views

Show: $M\subset\mathbb{R}^n$ Jordan-measurable, iff $vol^*(\partial A)=0$

Show that a bounded subset $A\subset\mathbb{R}^n$ is Jordan-measurable iff and only if $\partial A$ is a Jordan null set, i.e. $vol^*(\partial A)=0$. Here Show some properties of the ...
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1answer
54 views

Show that measure has a particular property

This is a general question: If I want to show that my measure has a particular property, how can I do this? For instance, my measure is invariant under orthogonal transformations of my sets, what are ...
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79 views

A detail on Lusin's theorem

Suppose that $B$ is a ball of $\mathbb{R}^{m}$, $(m\geq2)$, and $f(x)$ a measurable function on $B$. According to Lusin's theorem, we can find a closed set $F\subset B$ whose complement has a measure ...
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59 views

Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
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1answer
152 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
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33 views

Meaning of a probability on a product defined by a transition probability

If $(\Omega,\mathbb{B},P)$ and $(\Omega',\mathbb{B}')$ are sets with sigma algebras and $P$ is a probability on $(\Omega,\mathbb{B})$. If $\nu(\omega,B')$ is a transition probability from ...
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80 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
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2answers
41 views

Showing a set is measurable.

Let $X=Y=[0,1]$, equipped $X$ by giving Lebesgue measure on Borel sigma-algebra and equipped $Y$ by giving counting measure on the power set of $Y$. Define $D=\{(x,x):0\leq x\leq 1\}$ then how do we ...
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102 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
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43 views

One doubt regarding measurable function

Let $\phi(x,y)$ be a random variable on the product space $(R,B) \times (S,E)$ and $E|\phi(X,Y)| < \infty$ where $X$ and $Y$ are random vectors taking values in $(R,B)$ and $(S,E)$ respectively. ...
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50 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
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95 views

Birkhoff Ergodic Theorem Counterexample

I am trying to come up with a counterexample to this theorem under the assumption that the space is not sigma finite. I tried working with the power set of the real numbers with the measure $\mu(A) = ...
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1answer
238 views

How to prove that a Lipschitz function is absolutely continuous?

$f:[a,b] \rightarrow \mathbb{R}$ is a Lipschitz function. How to prove that it is absolutely continuous on $[a,b]$? My attempt: Let $\epsilon> 0$. Set $d = \epsilon/M$. If $P = \{[x_i, y_i]\}$ is ...
2
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1answer
83 views

X nonnegative and integral equal to zero implies measure of positive part zero

I'm trying to show that if we have X$\geq$0 and $\int{X} d\mu$=0, then that implies $\mu([X>0])$=0. I tried to do it using the definition $\int{X} d\mu$ = sup{$\int{Y} d\mu$: 0 $\leq$ Y $\leq$ X ...