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

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56 views

Purely nondeterministic weakly stationary processes

I found a necessary and sufficient condition for a stochastic process being purely nondeterministic in Ihara (1993). As follows: A weakly stationary process $X$ is purely non-deterministic if and ...
2
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1answer
38 views

Regularity of $\phi$ in order that $\int g_h \phi \,dx \to \phi(0)$

Define the sequence of functions $(g_h)_h$ where $$g_h(x):= h\, \chi_{[0,1/h]}(x)$$ and the sequence of measures $$(\mu_h(dx))_h:= g_h(x)\,dx.$$ We want to show that $\mu_h \stackrel{*}{\...
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1answer
24 views

Find two functions in $L_p(\Bbb R)$, whose product $f\cdot g$ does not belong to $L_p(\Bbb R)$.

How can I find two functions in $L_p(\Bbb R)$, with their product $f\cdot g$ not belonging to $L_p(\Bbb R)$?
2
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1answer
145 views

Natural and completed natural filtration not right-continuous

I'm looking for an example of a stochastic process, such that the natural filtration and the completed natural filtration aren't right-continuous. I defined the process $(Z_t)_{t \geq 0}$ as $Z_t = t ...
2
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1answer
60 views

The sigma algebra generated by open-dense subsets of $\mathbb{R}$

What is the description of the sigma algebra generated by all open-dense subsets of $\mathbb{R}$? Is it equal to the Borel sigma algebra?If not how is the structure of this sigma algebra?
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1answer
140 views

Two positive measures are mutually singular iff their sum is the variation of their difference

Let $\mu$ and $\nu$ be two finitely positive measures on measurable space $\left ( X, \mathfrak{A} \right )$. Prove that $$\mu \perp \nu \Leftrightarrow \mu + \nu = \left | \mu -\nu \right |$$ I ...
3
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1answer
48 views

Necessity of generalization of Dominated Convergence theorem

In Royden's Real Analysis there's this generalization of Lebesgue's Dominated Convergence theorem (p.92): Let $\{g_n\}$ be a sequence of integrable functions which converges a.e. to an ...
3
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0answers
38 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} u(x+rz)...
2
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1answer
61 views

Benefit from measure theory

With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral) What I know: With the Lebesgue integral we need less requirements to ...
3
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1answer
119 views

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: $$\int_{-\infty}^{+\infty}e^{it\lambda}\...
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0answers
31 views

Prove that $\lim_{r \to 1} \int_{-\pi}^{\pi} f(re^{i\theta}) d\theta = \int_{-\pi}^{\pi} f(e^{i\theta}) d\theta$

...if $f$ is continuous in an open set $\Omega$ containing the unit circle $T$. Is the proof something along the line of: $T$ is compact hence $\exists \epsilon > 0$ such that $D(z;\epsilon) \...
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0answers
119 views

Video lectures on Measure and Integration

Does anyone know a good online lecture series on measure theory and Lebesgue integration? I looked at the MIT open courseware but I could find only lecture notes. I am interested in lectures on this ...
2
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1answer
48 views

If $f(x) \le f(Tx)$ then $f(x)=f(Tx)$ almost everywhere ( $T$ is $\mu$-invariant )

Let $X$ be a probability space with probability $\mu$. Let $T:X\to X$ be a measurable and $\mu$-invariant transformation, i.e $\mu \left(T^{-1}A \right) =\mu A. $ for each measurable subset $A\subset ...
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0answers
86 views

Lebesgue measure of a parallelepiped

Suppose we have $n$ linearly independent vectors $\mathbf{x}_1$, $\cdots$, $\mathbf{x}_n$ in $\mathbb{R}^n$. Let $\mathbf{X}$ be the $n \times n$ matrix with column $k$ given by $\mathbf{x}_k$, $k = 1,...
3
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2answers
85 views

How to demostrate that $\int_{\{a\}} f(x) d \mu= \mu(\{a\}) f(a)$?

I don´t know how to demostrate that: $$\int_{\{a\}} f(x) d \mu= \mu(\{a\}) f(a)$$ Note: I have read on a book that "The Lebesgue Integral of a constant function on a measurable set will be that ...
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1answer
47 views

Basic measure theory question

It is a very important question for me. If we define the measure ${\mu}$ like this: ${\mu}=\left\lbrace \begin{array}{l} {\mu} (\lbrace a \rbrace )>0 \\ {\mu} (\mathbb{R}-{\lbrace a\...
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0answers
44 views

Borel regular measures equal if equal on open sets

Let $\mu$ and $\nu$ be Borel regular outer measures in a (separable metric) space X. We know that $\mu(U) = \nu(U)$ for every open set $U$. We wish to show that $\mu(A) = \nu(A)$ for every set $A \...
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2answers
58 views

Showing that the L1 norm of a given sequence of functions diverges

For $n=1,2,3,\ldots$ define $f_n:\mathbb R\to\mathbb R$ by $$f_n(x) = \frac{\sin(x)\sin(nx)}{x^2}.$$ Then certainly each $f_n$ is integrable on the real line. However, I have to show that the $L^1$ ...
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1answer
88 views

Jensen inequality for convex functions - infinite countable number of weights

Does Jensen inequality for convex functions hold if there is countable infinite number of weights? For example weights are given by sequence $(\frac{1}{2^n})_{n\in\mathbb{N}_1}$ ? If not, where is ...
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1answer
41 views

Itô's formula yields an Itô process

In our course on stochastic analysis, we proved the following version of the one-dimensional Itô formula: Let $\{W_t\}_{t\ge 0}$ be a one-dimensional Brownian motion w.r.t. some (right-continuous and ...
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1answer
23 views

Borel sigma field on inifnite (countable) set

If $\Omega$ is inifnite (countable) set, then borel sigma algebra on that set is equal to $2^\Omega$? What if $\Omega$ is uncountable?
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1answer
64 views

Linear Map is Homeomorphic in $\mathbb R^k$

I am having trouble understanding a proof in Rudin's "Real and Complex Analysis." The theorem states that To every linear transformation $T$ of $\mathbb R^k$ into $\mathbb R^k$ corresponds a real ...
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5answers
347 views

How can an open set be equal to a union of half-open sets?

In discussing $\sigma$-algebras I have seen the following used in proofs: $$(0,1)=\bigcup_{i=1}^\infty \left[ \frac{1}{n} , 1 \right)$$ In other words the open set is equal to a denumerable union of ...
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1answer
35 views

example for a function convergent in measure and in $L_p$ but not almost everywhere

I was looking for an example of a sequence $\{f_n\}_{n=1}^{\infty}$ such that $f_n\rightarrow f$ in measure and in $L_p$, but not almost everywhere. The book I'm studying (Real Analysis for Graduate ...
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1answer
426 views

Two sets $X,Y \subset [0,1]$ such that $X+Y=[0,2]$

A set $X\subset \mathbb{R}$ is called nice if, for every $\epsilon > 0$, there are a positive integer $k$ and some bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \...
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0answers
66 views

levy process and its characteristic function

Let $X(t)$ denote Levy Process. It can be proves that c.f of $X(t)$ is given: $E(e^{i\omega X(t)}) = e^{-\Phi(\omega)}$, where $ \Phi(\omega) = i \omega a - \int\limits^{-\infty}_{\infty} \frac{e^...
2
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1answer
47 views

Show limit exists of quotient of measures

This is a Theorem from Mattila's Book Geometry of sets and measures in Euclidean spaces: Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a separable metric space $X$. There ...
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1answer
47 views

Is $\partial (A\times B)$ jordan measurable when both of $A$ and $B$ are jordan measurable?

If $A\subseteq \mathbb{R}^{n} $ is Jordan measurable, $B\subseteq \mathbb{R}^{m} $ is Jordan measurable, then $A \times B \subseteq \mathbb{R}^{n+m}$ is Jordan measurable? We have $$\partial (A\...
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2answers
42 views

If $0 \leq f_n \leq g_n \rightarrow h$ in $L^2$ and $\int f_n^2 \rightarrow \int h^2$ then $f_n \rightarrow h$ in $L^2$

Let $(X,m)$ be a measure space, $(f_n)_n, (g_n)_n, h \in L^2(m)$. I would like to prove that if $0 \leq f_n \leq g_n$, $g_n \rightarrow h$ in $L^2$ and $\int f_n^2 \rightarrow \int h^2$ then $f_n \...
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1answer
125 views

Strong convergence of probability measures implies absolute continuity?

Suppose that $(\mu_n)_n$ is a sequence of probability measure for which $\lim_n\langle\phi,\mu_n\rangle=\langle\phi,\mu\rangle$ where $\mu$ is a probability measure and $\phi$ is any bounded, real ...
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1answer
29 views

Progressive measurability implies adaptedness

Somehow this statement in the title is obvious according to many textbooks but I couldn't produce a rigorous proof of it. Here is what I have so far. $(X_t)_{t\geq 0}$ being a stochastic process and $...
2
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1answer
67 views

No open set in $\mathbb{R}^n$ has measure zero in $\mathbb{R}^n$?

In section 11 of Munkres's Analysis on Manfiolds, question 2 asks you to prove that no open set in $\mathbb{R}^n$ has measure zero in $\mathbb{R}^n$, but isn't the empty set open and of measure zero ...
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1answer
28 views

On very basic Lebesgue integration

My understanding of Lebesgue integration is still lacking, so I'd like to start understanding better with a simple question: Let $f(x)=0$ if $x\in\mathbb{Q}$ and 1 otherwise. Let $\mu$ be a measure ...
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0answers
46 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
4
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1answer
148 views

Dual space of $L^{\infty}$ - Where is the mistake?

Today I thought about this for the first time and I really cannot see what is going on. I think it is a very stupid question but I really cannot see it. Consider the space $L^{\infty}(\mathbb{R})$ ...
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1answer
30 views

Integral over balls in $\mathbb{R}^n$ in different norms and measures

I need to calculate the integral $$ \int_{|x| \le r} f(x) \, dx $$ of a function in the $r$ ball in $\mathbb{R}^n$, using the standard Lebesgue measure. Take $f(x) = 1$, that is naturally the ...
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1answer
71 views

Extreme points of the set of positive regular borel measures on a compact Hausdorff space

I have some troubles with a specific proof of a (Bochner-type) theorem in Rudin's book "Functional Analysis". More specifically, let $X$ denote a compact Hausdorff-Space and let $M$ denote the set of ...
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0answers
38 views

Why is convergence of measures tested against functions?

This question is to help my intuition. Why do we test the convergence of measures against different classes of functions and not use definitions like: If $(B,\mathcal{B})$ is a measurable space then ...
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0answers
85 views

Measure of inverse image of points by an analytic mapping

How can one prove the following statement: For any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of $M$ ...
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1answer
46 views

What is the probability of 2 random matrices generate a free group?

Let A,B $\in GL(2,Z)$, then what is the probability of $<A,B>\cong F_2$? By probability, I mean the haar measure on $GL(2,Z)^2$. I already know what if we replace $\mathbb{Z}$ with $\mathbb{C}$,...
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1answer
117 views

Measure zero sets [closed]

Suppose that $E$ is a measurable set of real numbers with arbitrarily small periods. Explicitly, this means that there are positive numbers $p_{i}$, converging to $0$ as $i$ tends to infinity, so that ...
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1answer
46 views

On a proof of essential uniqueness of uniformly distributed measures

In the book 'Geometry of sets and measures in Euclidean space' by P. Mattila, theorem 3.4 states that Let $\mu$ and $\nu$ be uniformly distributed Borel regular measures on a seperable metric space $...
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150 views

Does $\pi$ contain infinitely many “zeros” in its decimal expansion?

Some number doesn't contain $"7"$ in its decimal expansion. For example Liouville's constant $$L=\sum_{n=1}^\infty\frac{1}{10^{n!}}=0.11000100....$$ contains only $0$ and $1$. It is well-known ...
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0answers
67 views

How to prove the sequence $f_n(x)=(\sin (nx))^n $ on the interval $[0,\pi]$ is not almost everywhere convergent?

How to prove the sequence $f_n(x)=(\sin (nx))^n$ on the interval $[0,\pi]$ is not almost everywhere convergent?
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0answers
17 views

sigma algebra generated by a collection of r.v

I am struggling to show the following: $\sigma(X_1,X_2,....X_n)=\sigma(\cup_{1,..n}\bf F_n)$ , where $\sigma(X_n)=\bf F_n$. Not sure how to express $\sigma(X_1,X_2,....X_n)$. Any help will be ...
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1answer
54 views

Set derived from definition of $\Vert f \Vert_\infty$

Someone told me that the set $B_n := \{x \in X : \vert f(x) \vert > \Vert f \Vert_\infty - \frac{1}{n}\}$ for $n \in \mathbb{N}$ (where $B_n$ has finite positive measure), is derived from the ...
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0answers
22 views

Random variables are independent iff transition probability is independent from first result $x$

I try to figure out why two real-valued random variables $X,Y$ (defined on $(\Omega, \mathcal A, P)$) are independent iff for all $B_1 \in \mathcal B^1$ the probability $P_1^2(x,B)$ is independent of $...
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1answer
43 views

Some questions on 'Almost All Subgroups of a Lie Group Are Free'

I am currently reading this paper:http://ac.els-cdn.com/0021869371901074/1-s2.0-0021869371901074-main.pdf?_tid=dbe2268a-086f-11e5-b2dc-00000aacb35d&acdnat=...
2
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1answer
52 views

Prove that a family of mollifiers converges in $L_\text{loc}^\infty$

Let $B_1$ be the open ball with radius $1$ around $0\in\mathbb{R}^n$ and $\phi:\mathbb{R}^n\to [0,\infty)$ with $\phi\in C_0^\infty(B_1)$, i.e. $\phi$ is infinitely many differentiable in $B_1$ and ...
0
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0answers
34 views

Problem on bounded variation functions

I need to give an example of a $\Bbb R \to \Bbb R$ continuous function compactly supported but not of bounded variation. I was thinking of $x\sin(1/x)$ kind of function. But exactly that does not ...