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

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Simulation of a random vector

I have a question which is probably well known but I do not find any written reference. Let us consider a probability measure $\mu$ on $\mathbb{R}^2$. I would like to know if one can find a random ...
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1answer
59 views

Haar Measure for Algebraic Number Theory: What Should I Know?

I recently taught myself some algebraic number theory and am preparing to take a course in class field theory this fall. I understand the notion of a Haar measure on a locally compact topological ...
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1answer
19 views

How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
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1answer
32 views

Metrization of the weak topology on the set of radon measures

Let $\mathcal{M}$ denote the set of Radon measures on $\mathbb{R}$. We endow $\mathcal{M}$ with the the weakest topology such that $\mu \to \int f \, \mathrm{d} \mu$ is continuous for all $f \in ...
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1answer
19 views

Return Lemma MC

If a Markov chain is $\phi$-irreducible and has stationary distribution $\pi$, then $\phi\ll \pi$, Proof: We use the irreducibility of the chain to write the state space $E = \bigcup_{n,m \in ...
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2answers
37 views

support of a probability measure

We know that if a probability measure is not continuous ,there will be a positive measure on a single point. must the quantity of these points be countable?
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1answer
69 views

Local integrability of $1/u$ implies $u>0$ almost everywhere

Let $U \subset \mathbb{R}$ be open. For a positive $\mathcal{B}(U)$-measurable function $u$, we define \begin{eqnarray*} S_{u}:=\left\{ x \in U: \int_{ \overline{B_{\epsilon}(x)}} u^{-1}(t)dt < ...
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2answers
90 views

Prove that a function with a measurable graph is differentiable [closed]

Let $f:[a,b]\to(0,\infty)$ be continuous and let $Gf:=\{(x,y)|y=f(x)\}$ be the graph of $f$. Prove that $Gf$ is measurable only if $f$ is differentiable in $(a,b)$?
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1answer
26 views

Integrability of the logarithm wrt a finite Borel measure

I have a finite Borel measure $d\phi$ on $(0,1)$, i.e. $\int_0^1 d\phi(x) < \infty$. Is it also true that $\int_0^1 \log (x) d\phi(x) < \infty$? The function $\log$ is integrable at 0, so ...
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1answer
33 views

Given continuity of measure, prove countable additivity to prove measure

I am trying to prove this theorem: Let ($\Omega$, $\mathfrak{F}$) be a measurable space. A mapping $\mu$: $\mathfrak{F}$ $\to$ $[0,\infty]$ is a measure iff 1 $\mu(\emptyset)=0$ 2 $\mu(A ...
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0answers
27 views

Searching for an Example

Let X be a set and A be an algebra over X. Also, let μ be a finite measure on A, and μ* be the corresponding outer measure, defined on all subsets of X. Can someone give an example of a subset Y of X, ...
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25 views

Space of measures is weak-* Hausdorff?

If $X$ is a topological space which is hereditarily Lindelöf and completely regular, then the space of finite signed measures on the Borel $\sigma$-algebra, endowed with the weak-* topology, is ...
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19 views

Fourier Transform for option pricing

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and sotck price Brownian motions of call options under stochastic interest rates? So lets ...
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1answer
33 views

Election measurable in uniform continuity

Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...
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1answer
51 views

Two notions of total variation norms

I found these two definitions of the total variation norm for probability measures on $(X,\mathcal{F})$: $$ \left \|\mu- \nu \right \|_{TV} = \sup_{\text{$f:X \rightarrow [-1,1]$ measurable}} \left ...
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1answer
71 views

Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit.

Let $f : [0,1] → \mathbb{R}$ be absolutely continuous, satisfy $f(0) = 0$ and $f′ ∈ L_2([0,1]).$ Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit. From absolute ...
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0answers
27 views

Why $\widetilde m = argmin_{m \in \mathcal P(X)} E[m]$ implies $\widetilde m (\arg \min \frac{\delta E}{\delta m}[\widetilde m]) =1$?

Consider $E: \mathcal P (X) \rightarrow \mathbb R \cup \{ \infty \}$ a functional (with a convex and dense domain, $E< +\infty$) over $\mathcal P(X)$ the set of probability measures of a metric ...
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1answer
42 views

If there exists an integrable function that is not zero a.e., then the measure is $\sigma$-finite

Suppose $f\in L^1(\Omega,\mathcal{A},\mu)$ and $f(x)\neq 0$ for almost every $x\in \Omega$. How to prove $\mu$ is $\sigma-$finite? I only got that $\Omega=\cup_{n=1}^\infty \{x\in \Omega:|f(x)|\geq ...
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1answer
65 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
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0answers
45 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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1answer
24 views

Proving that an ergodic and invariant map is constant a.e

I understand the first two sentences of the proof, however I cannot see how the third and final sentence holds. Why should $\mu(f \leq a)=0$ surely it should be non-zero as c is defined as the ...
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1answer
43 views

Using Kolmogorov's 0-1 law in proof of shift map being ergodic

Why should ${\cal E}_\theta$ be trivial?. I dont see how Kolmogorov's 0-1 law says that in this case we should take the 0 option. This is only mention of ${\cal E}_\theta$ in my notes I can find. ...
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1answer
116 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
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1answer
23 views

Integrability in a domain of $\mathbb{R}^{N}$

Hi everyone: Let $f$ be a function defined on an open set of $\mathbb{R}^{N}$ $(N\geq1)$. Is there any difference between the following two statements? 1) $f$ is locally integrable 2) $f$ admits a ...
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0answers
50 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
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0answers
32 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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0answers
17 views

Existence of Density in Bochner's Thoerem

Bochner theorem for locally compact abelian group, $G$ and a positive definite function $f$ there exist a unique measure $\mu_f$ such that: $$f(x)=\int\limits_{\hat G}(x,\gamma)d\mu_f(\gamma)$$ Where ...
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52 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
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26 views

Notation related to Markov kernels

We wish to jointly construct two copies $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ of a Markov chain on general state space, s.t. for $n=1,2,...$ $\mathcal{L}(X_{n+1}|X_n) = ...
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2answers
61 views

Limit of an integral, as the measure of the region of integration approaches zero

Hi everyone: Let $f$ be a function defined on on open set $D$ of $\mathbb{R}^{N}$, $(n\geq1)$. Suppose that $(\Omega_{\varepsilon})$ is a family of measurable sets in $D$ such that ...
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2answers
36 views

The set of $x$ where a sequence convergences in terms of set operations

I'm befuddled by this. Suppose $f:\mathbb{R}\to\mathbb{R}$, $f_n:\mathbb{R}\to\mathbb{R}$, $n=1,2,\dots$, and consider the set $$\bigcap_{k\geq 1}\bigcup_{p\geq 1}\bigcap_{m\geq p}\{x\in\mathbb{R} \ ...
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1answer
63 views

Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Let $F \subset \mathbb{R}$ be a closed set and define the distance from $x \in \mathbb{R}$ to $F$ by $d(x,F)= \inf_{y \in F} |x−y|.$ Prove that $$\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$$ for a.e. ...
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1answer
24 views

Proving result on measure's atoms

I have been told that a measure's atoms are at most a countable set. This has not been proved to me, and my book leaves it as an exercise. The only possible way I can think of is to try by ...
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1answer
28 views

Positive-definite function on a group function on a group

I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to ...
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1answer
27 views

Proving that the multiplication operator is closed

I was looking at this exercise: a) Let $f_n$ be a Cauchy sequence in $L^p(x,\mu)$. Prove that there exists a subsequence that converges pointwise $\mu$-almost everywhere. b) Let p $\in [1,\infty], ...
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1answer
40 views

Prove an identity without using Hölder's inequality

How to prove the following without using Hölder's inequality : $$ \|f\|_{p} = \sup_{\|g\|_q =1} \int |fg| d\mu ; \frac{1}{p} + \frac{1}{q} =1$$
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1answer
44 views

Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Let $(X,A,μ)$ be a finite measure space. Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$ a.) Prove that if $f_n → 0$ in measure, then $f_n → 0$ in $L_1(μ).$ b.) If ...
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25 views

Find Jordan decomposition constructed by function.

$$ F(x) =\left\{ \begin{array}{l l} 0 && x \leq 0\\ 1-x && 0 \leq x \leq 1\\ 2x-4 && 1 < x \leq 2 \\ 1 && 2 < x \end{array} \right. $$ ...
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1answer
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Does $u=0$ a.e. in $[0,T]\times\Omega$ imply for a.a. $t \in [0,T]$, $u(t)=0$ a.e. in $\Omega$?

Let $u:[0,T]\times\Omega \to \mathbb{R}$. Consider the statements $u=0$ a.e. in $[0,T]\times\Omega$ and for a.a. $t \in [0,T]$, $u(t)=0$ a.e. in $\Omega$ Does the first statement imply ...
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Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
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2answers
70 views

Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$

As part of an analysis qual problem, I am having a hard time showing that $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$. Any suggestions? Thanks in advance. I ...
3
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2answers
75 views

Best textbook for Geometric Measure Theory

I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure ...
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1answer
30 views

example concerning Lusin's theorem

Is there any example satisfying the following: $f$ is a measurable function on $\mathbb{R}^n$ with lebesgue measure $\lambda$. For any subset $N\subseteq\mathbb{R}^N$ with $\lambda(N)=0$, ...
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1answer
16 views

Vantage point tree question

I'm stuck in understanding the 1993 vantage point tree paper: http://aidblab.cse.iitm.ac.in/cs625/vptree.pdf It defines some things first: So if $x\in[0,1]$, then $P(x)$ is the probability of the ...
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1answer
26 views

What are the restrictions on infinite sums of measures?

Select a positive natural number at random. For all n, the probability of selecting that number n is zero. Yet the sum for all n of (probability of selecting number n) is 1. So there is some ...
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2answers
58 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
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1answer
27 views

Baire and Measure

Please,I would like help with this problem. We say that a subset $E$ of a topological space $X$ has the property of Baire if $E=G\Delta P$, where $G\subseteq X$ is open and $P \subseteq X$ is meager. ...
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1answer
41 views

Motivation for measurable functions?

Here is the definition for Measurable Functions we used this year in our Measure Theory module. Let $Z$ be a set and $\mathscr{A}$ be a $\sigma$-algebra of subsets of $Z$. A function $f:Z \to ...
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1answer
33 views

Counting measure proof

Can somebody tell me why the counting measure (so, if $S=P(X)$, then $\mu(A)$=infinity if $A$ isn't finite and $\mu(A)=$#$A$ if $A$ is finite) is a measure? (The second property of a measure isn't ...
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0answers
34 views

On $\sigma$-algebra generated by sets

Given $\mathcal{S}$ a collection of subsets of $X$ and $A\subset X$. To show that $\sigma(\mathcal{S}\cap A)=\sigma(S)\cap A$, where for any collection of $\mathcal C$ of subsets of $X$, $\mathcal ...