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

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3
votes
1answer
350 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
274 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
257 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
94 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
183 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
355 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
128 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
664 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
435 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
227 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 ...
4
votes
1answer
237 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
670 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
603 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
335 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 ...
1
vote
1answer
511 views

Borel measure of half-open and open intervals

the Borel set is the $\sigma$-ring generated by the open sets. One possible Borel measure on the real line is defined, for a closed interval, as: $$\mu([a,b])=b-a$$ But, from my understanding, ...
1
vote
1answer
136 views

Lebesgue measure and matrix notation problem

I have trouble with understanding following from my text book in Measures and Integral theory. Let T be an orthogonal $n\times n$ matrix. If $\lambda^{n}$ is the Lebesgue measure then we have: ...
16
votes
1answer
1k views

When is the image of a null set also null?

It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ ...
2
votes
1answer
357 views

$\sigma$ algebra of collection of random variables

Im doing a course on measure theory and I'm stuck on one of the exercises. Take $\{Y_{\gamma}:\gamma \in C\}$ as an arbitrary collection of random variables and $\{X_{n}: n \in N\}$ to be a countable ...
-1
votes
2answers
37 views

finding $f(g(x))$ and $g(f(x))$ in this problem

Suppose $f, g$ are two functions that $$f(x) = \begin {cases} 1 & |x|\leq 1 \\ 0 & |x|>1\end {cases}$$ and $$g(x) = \begin {cases} 2-x^2 & |x|\leq 2 \\ 2 & |x|>2\end {cases}$$ ...
0
votes
1answer
73 views

Measure induced by tanh

For a test prep question: Let $g(x)=\tanh (x)$, and let $\mu$ be the measure generated by $g$. Which subsets of the reals are $\mu$-measurable? Are polynomials integrable? And finally, is $|\sinh|$ ...
1
vote
1answer
115 views

Fatou's Lemma with $\max$

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$ so that $m(W)=1$. For all $k \in \mathbb{Z}_{\geq 0}$ let $f_k: X \times W \rightarrow \mathbb{R}_{\geq 0}$ be measurable and locally ...
4
votes
2answers
793 views

Composition of two absolute functions

$f$ and $g$ are two absolutely continous functions and $f$ is monotone. Is $f(g)$ necessarily an absolutely continuous function? Why not any counter example? Thanks a lot!!
3
votes
1answer
584 views

Limit Inf/Sup of Sequence of Set Example

In "A Probability Path", they have an example that states that the lim inf and lim sup of [0,n/(n+1)) is equal to [0,1). I guess I don't see how [0,1) is in all the sets except a finite number of ties ...
1
vote
0answers
461 views

complex integration over the whole plane

I am trying to solve this integral: $H(z)=\int_{\mathbb{C}}{p(z)\log_2{p(z)}dz}$, where $z$ is a complex number with complex normal distribution $p(z)$, and $\mathbb{C}$ denoted the complex plain. ...
2
votes
1answer
363 views

Question Related to Theorem that “Union of Two Measurable Sets is Measurable”

Let $\mu^* : \mathcal{P}(X) \rightarrow [0, \infty]$ be an outer measure, and let $M$ denote the set of $\mu^*$-measurable sets. Let $A \subseteq X$ and let $E,F \in M$. Why is the following ...
1
vote
2answers
179 views

Dominated Convergence Theorem for $p=\infty$?

Is there some version of DCT for $p=\infty$. That is, is it true that if there is a sequence of measurable functions defined on an open set $\Omega$ in $\mathbb{R}^n$, $f_n$ converging pointwise to a ...
4
votes
1answer
263 views

$L_p$ norm not subadditive for $0<p<1$ when endowed on $C[0,1]$

According to Wikipedia, the $L_p$-norm is not subadditive when $p\in(0,1)$. How can I show that the map $n_p(f)=(\int_0^1|f(x)|^p~\mathrm{d}x)^{2p}$ is not subadditive for $f\in C[0,1]$ for ...
4
votes
3answers
343 views

Lebesgue measure

How do I find the lebesgue measure of a interval $[n,n+\frac{1}{n^{2}}]$ when $n\in\mathbb{N}$? I have to use the following definition: The set-function $\lambda^{n}$ on ($\mathbb{R}^{n}, ...
1
vote
1answer
116 views

Relation among $L^{p}(\mathbb{R}^d)$?

Let $L^{p}(\mathbb{R}^d)$ be the linear space consists of $L^p$-integrable functions on $\mathbb{R}^d$ for $1\le p \le \infty$. Are there any relation among these spaces?
3
votes
2answers
390 views

semiring between measure theory and abstract algebra

what is the relation between semiring in measure theory and semiring in abstract algebra. why are they called the same? http://en.wikipedia.org/wiki/Semiring
2
votes
4answers
138 views

Proof: Tricky limit going to 0

I'm working on a proof and to complete it I need to find a way to choose an $n$ such that $(1-a)^n < \epsilon$ for a fixed $a$ such that $\frac12 < a < 1$ and any small $\epsilon$. I'm ...
2
votes
1answer
88 views

Notation for infinite product measure given marginal

I'm wondering what the preferred notation is for denoting a (countably) infinite product measure for which all marginals are equal to some given probability measure $\mu$. Is it common, for example, ...
1
vote
1answer
454 views

Under what conditions is expectation value distributive?

We know that for two real numbers $a,b$ and two random variables $X,Y$ we have that $E(a X + b Y ) = a E(X) + b E(Y)$. Under what conditions is it also true that for any three random variables ...
1
vote
1answer
140 views

$\sigma$-field generated by a r.v.

Let $(\Omega, {\cal F})$ and $(E, {\cal E})$ be measurable spaces and $X$ a r.v. from $\Omega$ into $E$ (i.e., $X$ is ${\cal F}/{\cal E}$-measurable). We assume that $\cal E$ is generated by ${\cal ...
0
votes
2answers
89 views

I have a measure theoretic proof which I think is incorrect.

To prove: Let $S$ be an arbitrary, non-empty set and let $\Sigma_0$ be an algebra on $S$. Let $\mu:\Sigma_0 \rightarrow [0,\infty]$ be a countably additive map. Show that for every decreasing ...
1
vote
0answers
308 views

Problem with the definition of Lebesgue Measure and borel sets!

The definition of lebesgue measure (in my textbook): The set-function $\lambda^{n}$ on ($\mathbb{R}^{n}, \mathcal{B}(\mathbb{R}^{n})$) that assigns every half-open $[[a,b)) = [a_{1},b_{1}) \times ...
1
vote
1answer
196 views

Having trouble with proofs using generated sigma algebras

I've just started with an advanced course in measure theory and I'm having trouble working with $\sigma$-algebras. Here's my problem: Let $(S,\Sigma, \mu)$ be a measure space. Call $N \subset S$ a ...
0
votes
1answer
216 views

Convergence in measure and pointwise convergence in continuity points

Hi can you help me with the following: $\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$. Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous. ...
2
votes
1answer
114 views

Characteristic functions (Statistics)

I would greatly appreciate any help with this problem. If $f_1, f_2 , f_3$ are three characteristic functions (in Statistics, e.g $E(\exp(itX)))$ such that $f_1*f_3=f_2*f_3$ for all $t$ and we are ...
6
votes
2answers
204 views

Is this actually true?

This exercise appeared in my real analysis test last year, and is still puzzling me since then. Ironically, even the professor doubts if the b part is actually truth (still...) Let $A \subset ...
1
vote
1answer
84 views

Is this true? Cardinality and subsets.

If you know that a given set O is countable. $\#O\leq \#\mathbb{N}$ Does this imply that the following statement holds? $\# O \leq \#\mathbb{R}$ I'm not sure, but I think it makes sense, ...
1
vote
4answers
108 views

What's $P(\{k\})$ in this exercise?

The question is based on the following problem in the book Probability Essentials by Jacod: Here is my question: What does $P(\{k\})$ mean in the second problem? The probability on a finite ...
5
votes
1answer
276 views

Uniqueness of Weak Limit

As we know that weak limit of a sequence of Borel probability measures on metric space is unique. Do we have this property for general sequence of signed Borel measures on metric space? Thank you.
5
votes
1answer
96 views

Inequality for Fourier transform of measure

I am having trouble with the following question. Let $\mu$ be finite measure on $\mathbb{R}$ and let $\hat{\mu}(\xi) = \int_{-\infty}^\infty e^{-ix \xi} d\mu(x)$ be its Fourier transform. Prove that ...
3
votes
2answers
587 views

Maps from sets of measure zero to sets of measure zero [duplicate]

Possible Duplicate: If $E$ has measure zero, then does $E^2$ have measure zero? I'm trying to find a proof for the following question: Suppose $A\subset\mathbb{R}$ is a set of measure zero. ...
3
votes
1answer
182 views

Atoms in a tail $\sigma$-algebra as $\liminf C_n$

Trying to solve exercise 1.1.18 in D.W. Stroock, Probability Theory, I somehow don't see how to get the hint in that exercise. Given a set $\Omega$, a tail $\sigma$-algebra $\tau$ generated by ...
2
votes
2answers
160 views

system of open intervals

Let $I$ be the set of all irrational points in $[0,1]$ and $\{J_n\}_1^N$ be a finite system of open intervals that cover $I$ . How to show that the $\sum_1^N \operatorname{length}(J_n) $ is greater or ...
0
votes
3answers
1k views

is the smallest $\sigma$-algebra containing all compact sets the Borel $\sigma$-algebra

Let $R$ be the smallest $\sigma$-algebra containing all compact sets in $\mathbb R^n$. I know that based on definition the minimal $\sigma$-algebra containing the closed (or open) sets is the Borel ...
2
votes
1answer
108 views

Convolution powers tend to zero

Let $\mu$ be a probability measure on $[0,\infty)$ that is not degenerated ($\mu(0) < 1$) and $f$ be a bounded function on $[0,\infty)$. Show that pointwise $$f * \mu^{*n} \rightarrow 0$$ where ...
4
votes
2answers
176 views

a problem on conditional independence

Given that $\mathcal{G}$ is a sub-sigma field. $Z=\mathbb E(X|\mathcal{G})$, how can we show that $X$ is independent of $\mathcal{G}$ given $Z$? I am struggling about the interpretation of this ...