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

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0
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1answer
32 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 ...
0
votes
5answers
273 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 ...
0
votes
1answer
26 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 ...
8
votes
1answer
298 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 ...
1
vote
0answers
43 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} ...
2
votes
1answer
41 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 ...
2
votes
1answer
36 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 ...
1
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2answers
31 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 ...
1
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1answer
27 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 ...
2
votes
1answer
17 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 ...
1
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1answer
47 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 ...
0
votes
1answer
22 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 ...
3
votes
0answers
26 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
votes
1answer
52 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 $L^{\infty}(\mathbb{R})$ ...
1
vote
1answer
18 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 ...
0
votes
1answer
46 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 ...
2
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0answers
32 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 ...
0
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0answers
71 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 $M$ ...
2
votes
1answer
41 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 ...
1
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1answer
83 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 ...
0
votes
1answer
24 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 ...
3
votes
0answers
68 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 ...
2
votes
0answers
40 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?
0
votes
0answers
13 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 ...
1
vote
1answer
44 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 ...
1
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0answers
14 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 ...
1
vote
1answer
38 views
2
votes
1answer
27 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
22 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 ...
3
votes
2answers
189 views

Finite function with infinite Lebesgue integral over any positive measure set

Is there a measurable function $f:\mathbb{R}\rightarrow [0,\infty)$ such that $$\int_A f\, \mathrm{d}\lambda=\infty$$ for any (measurable) set $A\subseteq\mathbb{R}$ with $\lambda(A)>0$. ...
3
votes
1answer
55 views

Bergman space. What is area measure?

I have read that the Bergman space $A^p(\Omega)$ consist of all the analytic functions $f$ in $\Omega$, such that $$ \left( \int_{\Omega} |f(z)|^p dA \right)^{1/p} < \infty $$ where $dA$ is the ...
4
votes
0answers
31 views

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk. My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , ...
2
votes
0answers
63 views

Is f(A) Lebesgue measurable when A is lebesgue measurable and f is a function of the class C1? [duplicate]

Let A be a Lebesgue measurable set. Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be a function of the class $C^1$; Is this true that f(A) is lebesgue measurable? I know that this is true when f is ...
0
votes
0answers
14 views

$\sum_{i=1}^{∞}\sum_{j=1}^{∞}a_{ij}=\sum_{j=1}^{∞}\sum_{i=1}^{∞}a_{ij}=\text{lim}_{r\to∞}\sum_{(i,j)\in rV}a_{ij}$

Let $a_{ij} ∈ \mathbb{C}$, $i, j = 1, 2, . . $ . If $\sum_{i=1}^{∞}\sum_{j=1}^{∞}|a_{ij}|<∞$ Then ...
1
vote
0answers
18 views

$(\int(\int f(x,y) dy)^p dx)^{\frac{1}{p}} \leq \int(\int f(x,y)^p dx)^{\frac{1}{p}} dy $

If $f: \mathbb{R}^2: \to \mathbb{R}$ measurable Lebesgue positive then for $ 1 ≤ p ≤ ∞$ $(\int(\int f(x,y) dy)^p dx)^{\frac{1}{p}} \leq \int(\int f(x,y)^p dx)^{\frac{1}{p}} dy $ . A suggestion to ...
0
votes
0answers
13 views

open sets tending in lebesgue measure to zero

Let $B_k \subset [0,1]$ be a sequence of open sets with the property that $\lambda(B_k)\rightarrow 0$ for $k\rightarrow \infty$, where $\lambda$ is the lebesgue measure. Further let $(\varepsilon_i) ...
0
votes
1answer
61 views

Filtration of Markov Chains in general state space

I am reading the book Markov Chains and Stochastic Stability from Meyn and Tweedie. They define Markov chains on a measurable state space $(E,\Sigma)$ (Chapter 3.4) and they define it on the space ...
0
votes
0answers
25 views

Logarithm of Probability measure of a set

What does the parameter K immediately suggest? Suppose we have a non-uniform probability measure $Q$ on a set of sequences of length $n$, $A$, and ${Q^n}$ is the corresponding product measure. $K = ...
0
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0answers
47 views

Regularity of measure on a locally compact Hausdorff space with a countable base

Let $X$ be a locally compact Hausdorff space with a countable base. Let $\mathcal B$ be the $\sigma$-algebra generated by the family of open sets of $X$. Let $\mathcal M$ be a $\sigma$-algebra ...
2
votes
0answers
27 views

Under what conditions on the experiment does bootstrapping work?

For a proof I would like to pretend that the uniform distribution on a finite set of samples from a 'source' eventually becomes the source's distribution a.s. when you keep adding samples. I am not ...
1
vote
2answers
58 views

Continuity of an integral

Let $f \in L^1(\Omega)$, where $\Omega$ is a bounded set in $\mathbb{R}^n$ Let $Z = (z_1,z_2,\dots,z_k)$ denote a k-tuple where each $z_i \in \Omega$ Consider $$F(Z) = \int_\Omega f(y)\min_{1\leq i ...
0
votes
1answer
24 views

Are continuous functions dense in $L^1$?

It is a well known fact that the continuous compactly supported functions are dense in $L^1(\mathbb R)$. An immediate counterexample to this fact for a non locally compact space is $\mathbb R ...
1
vote
1answer
25 views

Is the Lebesgue integral the same as the supremum of lower Darboux sums?

My textbook has a lot of definitions that look more or less the same thing to me so excuse my ignorance. It first defines a simple function as a function that can be written as ...
0
votes
1answer
22 views

$f_n$ converges in duality with $C_b$ and is uniformly bounded then it converges in duality with $L^1$

Let $(X,m)$ be a metric measure space, $(f_n)_n$ a sequence in $L^\infty, f \in L^\infty$ s.t. $$ \int gf_n \ dm \rightarrow \int g f $$ for every $g : X \rightarrow \mathbb R$ continuous and bounded. ...
0
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0answers
14 views

Differing conventions regarding “modulus character” of $k$-points of smooth affine $k$-group, $k$ non-Archimedean

Let $\mathbf{G}$ be a smooth connected affine $k$-group, where $k$ is a non-Archimedean local field, $G=\mathbf{G}(k)$ the group of $k$-rational points, a locally $k$-analytic group. Since $G$ is in ...
2
votes
1answer
61 views

Interchanging Inverse Laplace Transform

I have a function $f(|\boldsymbol{k}|,s,\theta)$ for which I am interested in its inverse Laplace transform. I am also interested in the function's mean value for constant $|\boldsymbol{k}|$, but ...
0
votes
0answers
48 views

Prove that $\mathbb{P}[(\limsup A_n)-A_n]\rightarrow 0$

Problem: This is a problem from Problems in Probability by Shiryaev (Problem 2.1.16). It is worded as following: Let $A^*=\limsup A_n$. Prove that $\mathbb{P}(A^* -A_n)\rightarrow 0$ Attempt: I tried ...
2
votes
2answers
42 views

Compactness and (global) convergence in measure

Let $B$ denote the unit ball of $L^\infty$. Question: is $B$ sequentially compact for the topology of convergence in measure ? I am not necessarily assuming that the measure is finite (but $\sigma$ ...
1
vote
1answer
19 views

Measurability of a Function Almost Equivalent to a Measurable Function.

I was asked during class to show the following statment: If $f$ is a measurable function and $g = f$ almost everywhere, then $g$ is measurable. This is simple to show under the Lebesgue measure (or ...
1
vote
1answer
24 views

Terminology : Hahn-Jordan decomposition

Here is what I could find in most analysis textbooks : Let $\mu$ be a signed measure on a measure space $(X, \mathcal{A})$. A Hahn decomposition is any pair $(P,N)$ of measurable sets such that $P ...