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

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0
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2answers
1k views

Finitely additive probability measure thats not countably subadditive

How is it that a finitely additive probability measure on a field may not be countably subadditive? I know that the field must be countably additive and thus finite additivity does not suffice, but ...
2
votes
1answer
127 views

Are $\mathscr{A}_\mu$ and $\mathscr{M}_{\mu^*}$ the same?

I am reading Cohn and he uses the notation $\mathscr{A}_\mu$ to mean the completion of $\mathscr{A}$ under $\mu$, and he says that a set in $\mathscr{A}_\mu$ is called $\mu$-measurable. He uses the ...
3
votes
3answers
164 views

Is a random variable bijective?

Given a probability space $(\Omega, \mathit{F}, \mathbb{P})$, a random variable is defined as a function $X: \Omega \rightarrow \mathbb{R}$ such that the set $\{ \omega: X(\omega) \leq c \}$ is ...
3
votes
1answer
174 views

Prove convergence without Lebesgue theory

W. Rudin has the following exercise, "to convince the reader of the power of Lebesgue integration". Let $0 \leq f_n \leq 1$ be continuous functions from $[0,1]$ to $\mathbb R$, such that they ...
2
votes
0answers
121 views

Additivity of integral, with a different definition of Lebesgue integration

I noted a definition of Lebesgue integral Here. This might be masochism; but how would one prove additivity? This is an "exercise for the reader" in the book of Lieb and Loss, "Analysis".
2
votes
2answers
369 views

Direct construction of Lebesgue measure

I have seen two books for measure theory, viz, Rudin's, and Lieb and Loss, "Analysis". Both use some kind of Riesz representation theorem machinery to construct Lebesgue measure. Is there a more ...
4
votes
1answer
302 views

Right continuous stochastic process

Can anyone suggest how to prove a right continuous stochastic process is measurable? Thanks Indrajit
4
votes
0answers
201 views

Computing the modularity function of upper triangular matrices

Put $B_p := \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \in GL_2(Q_p) : a, b, c \in Q_p \right\}$ the subgroup of upper triangular matrices in $GL_2(Q_p)$, $Q_p$ denoting the $p$-adic ...
2
votes
1answer
108 views

Show translation is not continuous in $\text{Lip}_\alpha(T)$

Let $f=\sqrt{|x|} \in \text{Lip}_\alpha(T)$, where $\text{Lip}_\alpha(T)$ is the set of Lipschitz function with Lipschitz constant $\alpha=1/2$ on the unit circle $T$. What is $$ \|f\|=\sup_{t\in T,h ...
0
votes
1answer
255 views

Not sure how to show that 2 generated sigma algebra are equivalent

I need to show that these generated $\sigma$-algebras are the same: $F_1 = \sigma(\{[a,b) : -\infty<a<b<\infty \}) $ $F_2 = \sigma(\{(-\infty , x] :x \in \mathbb{R} \})$ I am not sure, but ...
3
votes
1answer
235 views

Inner Measures of Subsets

How can I show that for $A \subset B$, it must be true that $m_*(A) \leq m_*(B)$? That is, the inner measure of A is less than or equal to the inner measure of B? I understand how to show a similar ...
4
votes
4answers
1k views

Open Measurable Sets Containing All Rational Numbers

So I am trying to figure out a proof for the following statement, but I'm not really sure how to go about it. The statement is: "Show that for every $\epsilon>0$, there exists an open set G in ...
5
votes
2answers
275 views

How to prove $\lim_{n\to\infty}f(nx)=0$ for almost every $x$, given $f:[0,\infty)\to[0,\infty)$ integrable.

I've tried an idea similar to this post. But that idea does not work here, since $\sum_{n=1}^\infty \frac{1}{n^2}<+\infty$ but here $\sum_{n=1}^\infty \frac{1}{n}$ does not converge. (This is ...
1
vote
0answers
84 views

Show with DCT: $\lim_{R \to \infty} \int_0^{2\pi} \frac{id\theta}{\sqrt{1+R^{-2}e^{-2i\theta}}}=\int_0^{2\pi} id\theta$

Let $R \in \mathbb C$. Is it possible to show $$ \lim_{R \to \infty} \int_0^{2\pi} \frac{id\theta}{\sqrt{1+R^{-2}e^{-2i\theta}}}=\int_0^{2\pi} id\theta $$ using the dominated convergence theorem?
4
votes
2answers
213 views

Do probability measures have to be the same if they agree on a generator of Borel $\sigma$–algebra $\mathcal{B}(\mathbb{R})$?

Suppose $\mathcal{K}\subset 2^\mathbb{R}$ is such that $\sigma(\mathcal{K})=\mathcal{B}(\mathbb{R})$ and let $\mu$ and $\nu$ be measures which agree on $\mathcal{K}$, i.e. $$\mu(A)=\nu(A)$$ for all ...
1
vote
1answer
87 views

Limit Volume of Parallel Sets

Given $F \subset \mathbb R^n$ non empty and $\epsilon > 0$. Let $F_\epsilon$ be $\epsilon$-parallel set of $F$, $$F_\epsilon := \{x \in \mathbb R^n:d(x,F)\le\epsilon\},$$ with $d(x,F):= \inf_{y\in ...
-2
votes
1answer
164 views

Radius of convergence of a series of random variables

Let $X_n$ be i.i.d. and (a.s.) bounded random variables.(none of them identically zero) Prove that the radius of convergence of the series with coefficients $X_n$, ...
0
votes
1answer
120 views

Is this map on a metric space upper-semi continuous?

Let $(X,d)$ be a metric space and $\mu$ a probability measure. Let $f(x)=\mu(B(x,r))$. Is this map upper semi continuous? I have some other assumptions since this is part of a larger proof but I'm ...
3
votes
1answer
681 views

The number of members of a sigma algebra generated by a set of finite number of subsets of $A$

Let $A$ be set of finite number of subsets of set $\Omega$. How many members are there in the sigma algebra generated by $A$ ?
0
votes
1answer
169 views

Probability measure on $\mathcal{P}(\mathbb{R})$

This question has been bugging me for a while? Does there exist a probability measure on the measurable space $\bigl(\mathbb{R},\mathcal{P}(\mathbb{R})\bigr)$. If so, what is it?
1
vote
2answers
263 views

If the measure on a measure space X is not $\sigma$-finite, does $X$ have infinite measure?

Let $X$ be a measure space with measure $\mu$. Suppose that $\mu$ is not $\sigma$-finite, so that $X$ is not the countable union of measurable sets of finite measure. Does this imply that the measure ...
4
votes
1answer
268 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
8
votes
2answers
1k views

Conditions that ensure that the boundary of an open set has measure zero

Are there some simple conditions which would ensure that the boundary of an open set in $\mathbb{R}^n$ has measure zero? Also, is it true that the boundary of a closed set in $\mathbb{R}^n$ has ...
3
votes
1answer
151 views

$\mu$ on $\mathcal{A}$ is $\sigma$ finite if and only if $\mu$ on $R$ is $\sigma$ finite

I have been struggling with the following problem for many hours now : Suppose $R$ is an algebra of sets on $X$ and $\mathcal{A}$ is the $\sigma$-algebra generated by $R$. Let $\mu$ be a measure ...
7
votes
2answers
376 views

constructing a sequence of simple functions with Lebesgue measure approaching the riemann integral

Let $\lambda$ denote the Lebesgue measure on the Borel sets of [0,1]. Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous. I know that the Riemann integral $I:=\int_{0}^{1} f(x)dx$ exists. I also know ...
1
vote
1answer
73 views

Boole and Chebychev inequalities on $(\Omega,\mathcal{B},P)$

Given a sequence of random variables $X_1,X_2,...$ defined on the same probability space $(\Omega,\mathcal{B},P)$. part 1: Verify that $P(\limsup_{n\to\infty}X_n>x)=0$ if and only if ...
7
votes
1answer
190 views

Why are measures real-valued?

Measures, as I understand it, exist to give us some sense of the "size" of a set. However, there's a lot of detail they gloss over, sometimes – the usual measure ignores all countable sets, for ...
2
votes
1answer
74 views

Set of Positive Measure Similar Triangle

If $A$ is a set of positive measure say in $\mathbb{R}^2$ then $A$ does not necessarily have a rectangle of positive measure. This is true I suppose? Because we can apply iteration in Cantor fashion ...
3
votes
2answers
140 views

Converse of f measurable $\Rightarrow \forall a, ${f=a} is measurable is false?

When is {f=a} measurable but f is not? Is this one of the times that including infinity is problematic? I'm trying to understand the definition of measurable function.
3
votes
1answer
1k views

Proving a function is Borel measurable

Prove the function $f:\mathbb{R}\rightarrow \mathbb{R} $ defined by $$f(x)= \begin{cases} 1/p, &\text{if $x=p/q$ is rational}\\ 0, &\text{if x is irrational}\\ \end{cases}$$ is Borel ...
3
votes
2answers
335 views

If $E$ has $\sigma$-finite measure, then $E$ is inner regular

In Rudin's Real & Complex Analysis, page 47: It is easy to see that if $E \in \mathfrak{M}$ and E has $\sigma$-finite measure, then $E$ is inner regular. $\mathfrak{M}$ in this context is ...
0
votes
1answer
81 views

No $E \in \mathcal{L}(\mathbb{R})$ such that $\alpha\lambda(I) \leq \lambda(E \cap I) \leq (1-\alpha)\lambda(I)$ for all intervals $I$.

For $\alpha \in (0,\frac{1}{2}]$, I am trying to show that there is no $E \in \mathcal{L}(\mathbb{R})$ such that for every interval $I$ we have $$ \alpha\lambda(I) \leq \lambda(E \cap I) \leq ...
1
vote
1answer
91 views

Show that $P=\sum_{n=1}^{\infty}2^{-n}P_n$ is a probability measure

Given: $P=\sum_{n=1}^{\infty}2^{-n}P_n$ I am trying to show that it is a probability measure, and further, that: $\int_{\Omega}XdP=\sum_{n=1}^{\infty}2^{-n}\int_{\Omega}XdP_n$ for any non-negative ...
8
votes
1answer
197 views

Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n $ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n $ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.
1
vote
1answer
135 views

Probability limit of a decreasing (uncountable) family of sets.

Let $t^*\in(0,1)$ and $(\Omega,\mathcal{F},P)$ be a probability space. Suppose I have set $A\in\mathcal{F}$ and an uncountable family of sets $(B_t : t\in[0,1])\subset\mathcal{F}$ with the following ...
0
votes
2answers
220 views

weak*-limit of bounded sequence of measures

Let $K$ be a compact Hausdorff space. Denote by $ca_r(K)$ the set of all countably additive, signed Borel measures which are regular and of bounded variation. Let $(\mu_n)_{n\in\mathbb{N}}\subset ...
3
votes
1answer
287 views

Does weak convergence of measures preserve absolute continuity?

Let $\{ \sigma_n \}$ be a sequence of positive measures on the complex unit circle $\mathbb{T}$ with its borel sets, and Suppose that $\{ \sigma_n \}$ converges weakly to $\sigma$ which is also such a ...
3
votes
2answers
260 views

Why Continuity set is a borel set?

$\def\R{\mathbb R}$Let $A= \{x: f \text{ is continuous at $x$}\}$ for $f : \R\to \R$ , why is $A$ Borel measurable?
10
votes
2answers
242 views

Just how continuous is measure

It's a classical theorem of real analysis that Lebesgue measure is "continuous" that is for an ascending chain of subsets $A_k$ we have $$\lim_{k\rightarrow \infty} m(A_k)=m\left(\bigcup_{k=1}^\infty ...
3
votes
1answer
92 views

Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.

I search the reference for the proof of the following theorem: Let $G$ be a locally compact group. Then the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable. The ...
3
votes
2answers
154 views

Does convergence of Fourier transforms imply convergence of measures?

Let $\{\sigma_n\}$ be a sequence of measures on the complex unit circle $\mathbb{T}$ and let $\sigma$ also be such a measure. Suppose that $\hat{\sigma_n}(k) \rightarrow \hat{\sigma}(k)$ as ...
1
vote
2answers
300 views

About an integral over measurable sets

Let $(X, \Sigma, \mu)$ a measurable space and $f$ an integrable function. Show that if $(F_n)_{n\in\mathbb N}$ is a decreasing sequence of measurable sets and $F=\bigcap_{n} F_n$, then ...
1
vote
1answer
127 views

Distinguish between Algebras

$$ S_n = \mathscr P (\{ -n, -n+1, \ldots, n-1, n\}) $$ $$ R_n = \{r : \Omega - r \in S_n\} $$ $$ T_n = S_n \cup R_n$$ I need to check whether $T_n$ is an algebra, semi-algebra or sigma algebra. ...
1
vote
1answer
546 views

Problem in proving a set is a Sigma Algebra

Let $(\Omega,\mathscr A)$ be a measurable space. If $\varnothing \subset X \subset \Omega$, let $$\mathscr F = \{ F \subseteq \Omega, F = X \cap Y, Y \in \mathscr A\} \;. $$ I need to prove that ...
2
votes
1answer
368 views

Is an elementary family of sets always an algebra of sets?

Definition 1: An algebra of sets on a non-empty set $X$ is a non-empty collection $\cal{A}$ of subsets of $X$ that is closed under taking complements and finite unions. Definition 2: An elementary ...
0
votes
1answer
199 views

Surface of spheres with the transformation formula

Let $\omega_n$ the measure of the surface (as obtained with the surface measure) of the unit sphere in $n$ dimensions. E.g. $\omega_2 = 2\pi$. Now let $n\ge 3$. I want to obtain a formula for the ...
3
votes
1answer
217 views

Relationship between two random variables?

What is the relationship between a random variable obeying the subexponential distribution defined here and a random variable $X$ satisfying $P\left(\left|X\right|>t\right)\le\alpha e^{-\beta t}$ ...
3
votes
2answers
622 views

Sigma Algebras generated by two classes of subsets

If $A_1$ and $A_2$ are two collection of subsets in $\Omega$ (Sample Space), I need to prove that $$\sigma(A_1) \subseteq \sigma(A_2).$$ I understand that there exist minimal unique ...
1
vote
1answer
557 views

Example that a measurable function $f$ on $[1,\infty )$ can be integrable when $\sum _{n=1}^{\infty }\int_{n}^{n+1}f$ diverges.

I am seeking help in my attempt to formulate a proof to disprove the following. For a measurable function $f$ on $[1,\infty )$ which is bounded on bounded sets, define $a_n= \int_{n}^{n+1}f$ for each ...
3
votes
2answers
285 views

Using LDCT to show a function is continuous and differentiable

We have the following test prep question, for a measure theory course: $\forall s\geq 0$, define $$F(s)=\int_0^\infty \frac{\sin(x)}{x}e^{-sx}\ dx.$$ a) Show that, for $s>0$, $F$ is ...