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

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Prove absolute continuity without Banach-Zarecki

Let $f$ be a real-valued continuous function of bounded variation on $[a,b]$. Suppose $f$ is absolutely continuous on $[a+\eta,b]$ for every $\eta\in(0,b-a)$. Show that $f$ is absolutely continuous on ...
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1answer
40 views

Does a difference of variables generate the same Sigma-algebra?

When reading the textbook Probability and Measure, I found the below part, Note that, since $X_k=\Delta_1+\cdots+\Delta_k$ and $\Delta_k=X_k-X_{k-1}$, the sets $X_1,\ldots, X_n$ and $\Delta_1,\...
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11 views

Why is a function $g: I \to \mathbb{R}$ that is increasing and right continuous a CDF of a unique Borel measure?

Given a closed bounded interval $I$, why is a function $g: I \to \mathbb{R}$ that is increasing and right continuous a cumulative distribution function of a unique borel measure?
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1answer
78 views

conditional probabilities on densities

I have a seemingly basic question, but surprisingly my web search didn't give any satisfying answers. Let $F(s)$ be the distribution of some random variable $X$ of support $(a,b)$ with continuous ...
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0answers
11 views

Why is a risk neutral measure unique in a discrete time market with continuous states?

Why is the radon nikodym derivative unique in a discrete time market with continuous states? By radon nikodym derivative, I meant the derivative with respective to the risk neutral measure and the ...
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0answers
17 views

For showing measurability of Brownian motion, how does this set equality holds?

It is stated that the the following set equality easily comes from continuity of paths of Brownian motion $B_t$, but I can't seem to make sense of it - $$\{(\omega,t)\in \Omega\times (0,\infty) : ...
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103 views

Halmos Measure Thoery section 62 exercise 3

Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed ...
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2answers
112 views

For $\phi(x,y)=(x,y+\psi (x))$ with $\psi:\Bbb{R}\to \Bbb{R}$ integrable, show that $\phi (B)$ is measurable for every box $B\subset \Bbb{R}^2$

Let $\psi:\Bbb{R}\to \Bbb{R}$ be integrable and define $\phi:\Bbb{R}^2\to \Bbb{R}^2$ by $\phi(x,y)=(x,y+\psi (x))$. Prove that for every box $B\subset \Bbb{R}^2$, $\phi(B)$ is measureable and $v(\phi (...
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3answers
52 views

Lebesgue measure of a sphere

While reading proofs (for ex. this) about measure theory I am inclined to think that it is implicitly intended that the $n$-dimensional Lebesgue measure of a hypersphere $\mathbf{S}^{n-1}$, i.e. of ...
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1answer
66 views

Topology and Borel sets of extended real line

Let $\mathcal{B}_{X}$ denote the Borel $\sigma$-algebra on $X$. I'm reading a book on real analysis by Folland and he defines $$\mathcal{B}_{\overline{\mathbb{R}}} = \{ E \mid E \cap \mathbb{R} \in \...
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17 views

Measure extension

Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},P)$ with convention $\mathcal{F}=\bigcup_{t\geq 0}\mathcal{F}_t$. Given a positive $(P,\mathcal{F}_t)$-Martignale $M_{...
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7 views

How to calculate sigma algebra generated by random variables in stochastic process?

According to http://www.math.uah.edu/stat/processes/Stop.html, a stochastic process $X=\{X_t:t \in T\}$ is a stochastic process with state space $(S, \mathscr{S})$ defined on underlying probability ...
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75 views

Moments of positive Carleson measure

A result of Widom from the 60's shows that a measure $\mu$ on the unit disc $\mathbb D$ concentrated on $(-1,1)$ is a Carleson measure if and only if $$ \int_{(-1,1)} t^k\,d\mu(t) = O(1/k)\quad\text{ ...
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0answers
44 views

Is $\sin^\alpha(x)$ absolutely continuous on $[0,1]$ for $\alpha\in (0,1)$?

Prove or disprove: $f(x)=\sin^{\alpha}x$ is absolutely continuous on $[0,1]$ for all $\alpha\in(0,1)$. Here is my thought process: I know that $f$ is absolutely continuous on $[0,1]$ if and only if ...
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1answer
40 views

If $f \in L^1([a, 1])$ for all $a \in (0, 1)$, is it true that $f \in L^1((0, 1])$?

I'm learning about measure theory and need help with the following question: True or Fasle (justify): If $f \in L^1([a, 1])$ for all $a \in (0, 1)$, then $f \in L^1((0, 1])$. While it is very ...
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36 views

Lebesgue Integral vs. Lebesgue Stieltjes Integral

Forgive me if this has been addressed in a question already on here (and for my lack of comfort with measure theory), but is there any difference between the Lebesgue integral and the Lebesgue-...
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1answer
41 views

$\limsup_{k\rightarrow\infty}X_k/k$ for identically distributed random variables $X_k$.

Let $\{X_k\}_{k\in\mathbb{N}}$ be a sequence of identically distributed and not independent random variables on the natural numbers. I am interested in conditions that would ensure that almost surely $...
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0answers
16 views

About measures which respect half-spaces

I was wondering if there are measures known on $\mathbb{R}^n$ which somehow "nicely" see the half-spaces. I am not sure how to exactly quantify "nicely" and hence feel free to make your own ...
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2answers
120 views

Characterization of measurable functions

Let $X$ be an uncountable set, let $\mathfrak{M}$ be the collection of all sets $E\subset X$ such that either $E$ or $E^c$ is at most countable, and define $\mu(E) = 0$ in the first case, $\mu(E) = 1$ ...
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1answer
46 views

Can every function $f$ be written as the limit of simple functions?

Actually, I have ever posted about this theorem. However, the reason why I asked this is I want to make sure about this theorem. In my textbook referred, (I copied the theorem exactly same as the ...
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2answers
121 views

Computing $\int_0^{\infty}\frac{\cos(x)-1}{x^{1+\alpha}}\,\mathrm d x$

Let $\alpha\in(0,1)\cup(1,2)$. I want to show that the integral $$\int_0^{\infty}\frac{\cos(x)-1}{x^{1+\alpha}}\,\mathrm dx$$ exists (in Lebesgue’s sense: the integral of the absolute value of the ...
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1answer
41 views

Kolmogorov's Truncation Lemma (iii)

Probability with Martingales: In the definition of $f$, is that really $z$ and not $\lceil |z| \rceil$, $\lfloor |z| \rfloor + 1$ or something? How exactly do we have the part in the $\...
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1answer
70 views

Kolmogorov's Truncation Lemma (ii)

Probability with Martingales: How exactly do we have the part in the $\color{red}{\text{red}}$ box? What I tried: $$E\left[ \sum_{n=1}^{\infty} 1_{|X| > n} \right]$$ $$ = E\left[ \...
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2answers
79 views

Strong Law of Large Numbers - Converse

Probability with Martingales: I want to try to show the last one $$\left[\limsup \frac{|S_n|}{n}\right] = \infty \ \text{a.s.}$$ which is equivalent to $$\forall k \in \mathbb N$$ $$\left[\...
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1answer
29 views

If $\{f_n\}$ converges pointwise to $f$, and $\{f_n\}$ and f belong to $L^p(E)$, why is $|\|f_n\|_p-\|f\|_p|\le \|f_n-f\|_p$?

If $\{f_n\}$ converges pointwise to $f$, and $\{f_n\}$ and $f$ belongs to $L^p(E)$, why is $$|\|f_n\|_p-\|f\|_p|\le \|f_n-f\|_p \text{ ?}$$
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16 views

Showing that Polya's Urn is a martingale - is this correct definition of Polya's urn?

Possibly related: Show rigorously that Pólya urn describes a martingale Suppose we had $b$ and $r$ blue/red balls in urn at time $0$, and at each $n\ge 1$ we draw a ball randomly and then put it ...
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1answer
12 views

Does countably monotone imply finitely monotone and vice versa?

If a set function $\mu$ is finitely monotone and has the property that $\mu(\phi)=0$, does it imply it is countably monotone? Royden claims that if a countably monotone function $\mu$ has the ...
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1answer
67 views

Not integrable although iterated integrals are equal

How can I show that the function $$f=\begin{cases} 0 & (x,y)=(0,0)\\\frac{xy}{(x^2+y^2)^2} & \mbox{else}\end{cases}$$ is not Lebesgue-integrable, although the iterated integrals exist and are ...
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1answer
31 views

Three-dimensional Lebesgue-measure

How can I compute $\int_B gd\lambda^3$ where $$g(x,y,z)=xyz$$ and $$B=\{(x,y,z)\in\mathbb R^3\vert x,y,z ≥ 0, x^2+y^2+z^2 ≤ R^2\},$$ $R>0$ arbitrary? I have no clue on how to find the upper and ...
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1answer
32 views

Doubt on proof of equivalance of conditions for uniform integrability in $L^1$

In a measure space with finite total measure, a family $A$ of r.v's is called U.I if $$ \lim_{N\to\infty}\sup_{X\in A} \int_{|X|>N} |X|\,d\mu=0 $$ I have some doubts on equivalance of this ...
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56 views

$\mu * \nu$ a finite Borel measure in $\mathbb{R}$?

Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb{R}$. For any Borel set $A \subset \mathbb{R}$, define$$\mu * \nu(A) = \mu \times \nu(\{(x, y) \in \mathbb{R}^2 : x + y \in A\}).$$Is $\mu * ...
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24 views

Can anyone help me understand one step in the constructing of product measure?

Given two measure space (X, A, $\mu$), (Y, B, v), let $\{A_k * B_k\}_{k=1}^{k=\infty}$ be a countable disjoint collection of measurable rectangles whose union also is a measurable rectangle A * B. ...
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29 views

CLT for continuous functions of random variables

Let $(X_i)$ be a collection of zero mean, unit variance, real valued random variables (I do not assume that they are iid). Let $\mathcal H$ be a separable RKHS with a bounded kernel $k(x,y)$. Suppose ...
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32 views

Why is Caratheodory's characterization of measurability important?

My professor repeatedly emphasizes the importance of Caratheodory's theorem about characterization of measurability, but I don't get why it's so important. As far as I remember, I have never used this ...
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1answer
31 views

Algebra on infinite sets is not closed under countable union

this proof came up during self study, rather than looking at the other proofs out there, I would like to correct if necessary the following one... Let $X$ be an infinite set and $A$ the collection of ...
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1answer
40 views

An uncountable chain of equivalence relations

First, an example: We know that, for two real valued, Lebesgue-integrable functions, the relation "equals almost everywhere" is an equivalence relation. In particular, if $f_0$ is Lebesgue-integrable, ...
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3answers
26 views

Calculating $E[X \vert f(X) \leq c]$

Let $X$ be distributed uniformly on $[0,2]$ and $f(X) = \beta X$, $0 < \beta <1$. Also, let $f(0) = 0$ and $f(X) < X$ for $X>0$. Note that $c,\beta$ are constants. Then: $$ E[X \vert f(X) \...
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1answer
45 views

A Measure Theory problem-If $\int_{A_n}f(x)dx\rightarrow0 $ then $\lambda(A_n)\rightarrow0$

This question was proposed as part of a test for PhD applicants but considered too hard and rejected. I tried unsucessfully to solve it for quite some time. For anyone wishing to try his luck.. ...
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37 views

Subspace of $L^1(\Omega)$ closed

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ two sub-sigma-algebras and $f$ a $\mathcal{B}$-measurable function. I want to show, that the subspace $$\...
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2answers
22 views

Natural filtration of martingales

I don't quite understand what the natural filtration really is. Imagine e.g. a sequence of independent and identically distributed random $N(0,1)$ variables. What is their natural filtration, and how ...
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1answer
34 views

Measure theory: upper bound for a particular set

I have the following problem: consider $A_1,...,A_N$ Borel set on [0,1] with measure greater than 1/2. For every a real number between $0$ and $\frac{1}{2}$, consider the set $E_a$={$x$| $x\in A_j$ ...
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22 views

Problem 1.3 from RCA Rudin

Prove that if $f$ is a real function on a measurable space $X$ such that $\{x : f(x) \geqslant r\}$ is measurable for every rational $r$, then $f$ is measurable. Proof: For every $\alpha\in \mathbb{R}...
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25 views

Situation where the conditions of Kolmogorov Consistency Theorem not hold [closed]

I'm wondering what is a possible finite dimensional distribution that violates the two conditions in the Kolmogorov extension theorem. It's hard for me to imagine what distribution violates these ...
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1answer
19 views

$\sigma$-finite versus Locally Finite Measures

Which implications are true (if any) for a measure $\mu$: $\sigma$- finite $\implies$ locally finite locally finite $\implies$ $\sigma$-finite My guess would be that both are false, but ...
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35 views

An example of outer measure.

First a few definitions: 1.5.1: Definition. Suppose that $\mu$ is a nonnegative set function on domain $\mathcal{A} \subset 2^X$. A set $A$ is called $\mu$-measurable if for any $\epsilon>0$, ...
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14 views

Bayesian equation: need for priors

As far as I understand, in the problem of Bayesian inference we have a random variable $y$ describing data, which is distributed according to some parameter $x$ via the known conditional distribution $...
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1answer
43 views

Is $\iint \dfrac{1}{z} dxdy\neq 0$?

I am trying to solve an exercise and at some point I came accross the integral $$\iint_L \dfrac{1}{z} dxdy,$$ ($z=x+iy$) where $L\subset \mathbb{C}$ is a compact set with positive two-dimensional ...
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34 views

Is Bayesian Association mathematically rigorous?

Introduction. This question is based on the Ph.D. thesis of B.T. Vo, which can be found in this website ("Papers" section). More specifically, in the introduction of the Ph.D. thesis, at page 8, there ...
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14 views

Function in indicator function and integral.

I have the following expression $$\int 1_{ t^{-1}(A)}(x) e^{t(x)} d\mu(x)$$ How do I express this integral in terms of $t(\mu)$? Specifically, what do I do about that indicator function which also ...
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2answers
31 views

Determining null sets with Tonelli's theorem

How can I show that the diagonal $D=\{(x,y)\in\mathbb R^2\vert x=y\}$ is a Lebesgue-nullset in $\mathbb R^2$ by utilizing the theorem of Tonelli? My solution so far, but it doesn't seem quite right: ...