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

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3
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3answers
68 views

Existence of a countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set. If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite. If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and ...
0
votes
0answers
10 views

What is a normalized measure?

Let $\Omega$ be a compact metric space and $\gamma$ a normalized $\mathbb{R}$-invariant measure on $\Omega$. I onder what is meant with normalized measure and R-invariant. Do you have an ...
0
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0answers
25 views

Sigma algebra generated by a random vector

I understand this question is very basic, but I found this confusing while I am learning measure theory myself.. Suppose we toss a coin twice (once afeter once), and denote by each $X$ and $Y$ the ...
2
votes
1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
0
votes
0answers
18 views

Showing a fact about $\sigma$-algebras and Borel sets

Let $(\Omega,\mathcal{A})$ be a measurable space, $(A_n)_{n\in\mathbb{N}}\subset\mathcal{A}$ and $f_n:\Omega\to [-\infty,\infty]$ be a $\mathcal{A}-\overline{\mathcal{B}}$ measurable function, where ...
2
votes
1answer
22 views

Predictable process with stopping time

I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it. Let T be a stopping time in $\mathcal{F}_t$ for ...
-1
votes
1answer
47 views

Borel Measures: Discrete & Continuous? [on hold]

Here, the focus lies on discrete & continuous - not atomic & atomless!!! What is the rigorous definition for a Borel measure to be continuous? (The definition for discrete measure can be ...
0
votes
1answer
21 views

A function on set involved in product of measurable sets

Let $\mathfrak{S}_1$ and $\mathfrak{S}_2$ be two families of measurable sets, and let $C\in\mathfrak{S}_1\times\mathfrak{S}_2$ be the countable union of disjoint sets, i.e. $C=\bigcup_{n=1}^\infty ...
1
vote
2answers
45 views

$L^{p}$ spaces and their properties

I have aquestion :Idont know how to show that if $1<p<q<\infty$ , then $L^{q}$(0,1)$\subset$$L^{p}$(0,1) and $\mid\mid f\mid\mid$$_p$ < $\mid\mid f\mid\mid$$_q$ ,f $\in$$L^{q}$(0,1)? ...
2
votes
1answer
28 views

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k ...
1
vote
3answers
27 views

Understanding sigma super additivity

An additive measure $\mu$ on $R$ has the property that for pairwise disjunct $A_i \in R$ with $\left(\bigcup_{n=1}^\infty A_i\right)\in R$: $$ \mu\left(\bigcup_{n=1}^\infty A_i\right) \geq ...
3
votes
2answers
32 views

Probability Assignment to Intervals in $\mathbb{R}^{n}$.

Given a random variable $\bf{X}$ distributed on $\mathbb{R}^{n}$, let $F_{X}(t)$ be its distribution function. Suppose we want to find $P\left(\textbf{X} \in (\textbf{a}, \textbf{b}]\right)$. I was ...
0
votes
0answers
33 views

Find a line with measure 0

A finite measure $m$ is defined on a $k$-connected set $D$, with $k>1$. You want to convert $D$ into a $(k-1)$-connected set without hurting the measure. Formally: prove that there is a set ...
1
vote
1answer
16 views

Application of existence and uniqueness of measure

I have the following situation: $\Omega$ is a finite set, $A$ a sigma-algebra on it. Suppose I have a finite partition of $\Omega$ into pairwise disjoint subsets $\pi_i$, such that I can write every ...
1
vote
1answer
22 views

What does $\lVert\mu \rVert=1$ mean for a measure $\mu$ on a compact metric space?

What does $\lVert\mu \rVert=1$ mean for a measure $\mu$ on a compact metric space $\Omega$? Sorry, I would like to add some own ideas, but I do not have... it has to be a kind of normalization on ...
0
votes
1answer
16 views

Ergodic theory for flows: Invariant

Here is the definition of an invariant measure that I do know: Let $(X,\Sigma)$ be a measurable space and let $f\colon X\to X$ be measurable. A measure $\mu$ on $(X,\Sigma)$ is saif to be ...
1
vote
1answer
21 views

Is an infinite product probability space compatible with an almost surely statement?

My question pertains to the following Theorem which can be found here. Theorem (Existence of product measures): Let $A$ be an arbitrary set and for each $\alpha \in A$ let ...
0
votes
0answers
44 views

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? [duplicate]

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? This is not a homework problem but rather a question I had. If it is not true, what are the ...
1
vote
1answer
42 views

$\int f(x) dm = \int f(x + a) dm$

$f: \mathbb{R} \rightarrow [0,\infty]$ is Lebesgue-measurable and $m$ is the Lebesgue measure. Show that for $a\in \mathbb{R}$ $$ \int f(x)dm = \int f(x+a) dm.$$ I know Lebesgue measure is ...
1
vote
1answer
11 views

Support, measure, function

I have a question about a support. Let $m$ be Lebesgue measure on $\mathbb{R}^{d}$ and $f$ be a continuous function on $\mathbb{R}^{d}$. We define $\mu(A):=\int_{A}|f|dm,\,A{\rm \,: ...
1
vote
1answer
44 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
0
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0answers
15 views

Total Variation of a Signed Measure

In Royden's Real Analysis the total variation $|\mu|$ of a signed measure $\mu$ is defined to be $$|\mu|(E) := \sup\sum_{k=1}^n |\mu(E_k)|,$$ where the supremum is taken over all finite disjoint ...
0
votes
1answer
9 views

A property of conditional expectation

Given a probability space $(X , \mathcal{M} , m)$ and $\mathcal{A}$ is a $\sigma$ sub algebra of $\mathcal{M}$. Let $\mathbb{E}$ be the condition expectation given $\mathcal{A}$. Given $f$ is an ...
0
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0answers
12 views

Conditional expectation, conditioning on a random variable [duplicate]

I am asked to show that if $X,Y$ are integrable, and $E[X| Y]=Y$ and $E[Y|X]=X$ almost surely, then $X=Y$ almost surely. Moreover, is the first equality sufficient for $X=Y$ almost surely? My attempt ...
0
votes
1answer
44 views

borel measurable and measurable

Let $(\Omega,A,\mu)$ be a measure space and let f:$(\Omega,A)$$\to$$(\mathbb{R},\mathbf{B})$ be a nonnegative measurable function and define define the graph ...
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votes
0answers
15 views

Monotone measureable function that is bounded on domain [on hold]

Given $( \mathbb X , \mathscr A, u)$, Given a measurable, monotone function $f$ defined on a bounded set, Show that this function is $u$-integrable.
2
votes
0answers
57 views

Measure-preserving maps on probability spaces

A professor posed me a problem a few days ago, and I have not been able to find an answer to it. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following ...
2
votes
1answer
21 views

Relationship between $m(E\backslash F) < \epsilon$ and $m(E)-m(F) < \epsilon$

This may be a fairly easy question but I just want to make sure there are no strange counterexamples. Suppose $E$ and $F$ are measurable sets such that $F \subset E$. Now suppose $m(E \setminus F) ...
0
votes
0answers
26 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
0
votes
0answers
28 views

Tonelli-Fubini Theorem for two copies of $\mathbb{N}$ with counting measure

I just learned the Tonelli-Fubini Theorem and I was wondering what does it say for two copies of $\mathbb{N}$ when considering the counting measure. And what is the difference when we consider one ...
0
votes
1answer
49 views

Show that the measure is zero

I am asked to show that the $2-$dimensional Lebesgue measure of the graph of a continuous real function is zero. Could you give me some hints how I could show it??
1
vote
0answers
22 views

Functional representation of adapted jointly measurable stochastic processes

Let $X_t : \Omega \to E, \ t \geq 0$ be continuous-time stochastic process with (Polish) state space $E$ and canonical filtration $\mathcal{F}_t := \sigma(X_u \ | \ u \leq t)$. Let $Y_t : \Omega \to ...
0
votes
0answers
8 views

Continuity of signed measure

I want to prove that, given a signed measure $\nu$ on $(X,M)$, continuity from above and from below continues to hold. What I'm uncertain about is any difference in the proof for a signed measure ...
2
votes
1answer
29 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
0
votes
1answer
19 views

Sigma-algebra generated by a sigma-algebra

I know intuitively that $$\sigma(\sigma(\theta))=\sigma(\theta),$$ where $\theta$ is the class of all open sets in $\mathbb{R}^\mathrm{k}$. But why? How can I prove it? Also, is ...
0
votes
0answers
19 views

Polynomial density in $L^p (\mathbb{R},\mu)$

I wanna check a necessary and sufficient condition for a Radon measure witch have the moments of all orders, to say that polynomials are dense in $L^p (\mathbb{R},\mu)$. Or just a paper or an article ...
1
vote
1answer
58 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
2
votes
0answers
28 views

How to show function is measurable if and only if each component is

Let $(X,\mathcal{M})$ be a measurable space and $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ be a measurable space of Borel sets on the real line. Let $f_i:X\rightarrow \mathbb{R}$ be given for ...
0
votes
1answer
15 views

Lebesgue measurable functions

Let $f(x)=x^{-1/2}$ if $0<x<1$ and $f(x)=0$ otherwise. Is f a Lebesgue measurable function from $\mathbb{R} \rightarrow [0,\infty]$? If it is a Lebesgue measurable function how can I show it?
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votes
1answer
17 views

Different ternary representations

I just picked up the subject of ternary expansions (actually I'm trying to gain an understanding of the cantor set for measure theory) so my knowledge is still extremely weak but I just a quick ...
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0answers
22 views

On Compact and Measurable Sets with Positive Length

Greetings fellow Mathematics enthusiasts! The following two-part problem is giving me trouble, and I was hoping someone could help me solve it. It is coming from Terrence Tao's Introduction to Measure ...
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0answers
9 views

Showing that a precursor to the packing measure is not a measure

I am trying to prove the highlighted sentence. What countable dense sets should I consider? and how am I trying to prove this is not a measure?
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0answers
63 views

Complete additivity of set functions

I need help with the following two statements of measure theory, for which I am learning for at the moment: For any non-empty set $T \subseteq [0, \infty]$ of non-negative numbers (allowing ...
2
votes
1answer
23 views

Borel measure and Riesz measure

Let $\mu$ be a Borel measure on $\mathbb{R}^n$ s.t. $\mu(K)< \infty$ for all compact $K$. Show that $\mu$ is the restriction of some Riesz measure on $\mathcal{B}$. I try to prove it using the ...
0
votes
0answers
32 views

Characterization of Lebesgue measure based on translation invariant

I am trying to solve a problem about characterization of Lebesgue measure. Let $(\mathbb{R}^n, \mathcal{B}, \mu)$ be a Borel measure space whose measure $u$ is translation invariant and exist a set ...
0
votes
3answers
79 views

Prove that $\|x+y\|_{\infty} \leq \|x\|_{\infty} + \|y\|_{\infty}$.

Suppose $\left(X, \Sigma, \mu \right)$ is a measure space and $x,y \colon X \longrightarrow \mathbb{R}$ are random variables. We define $$\|x\|_{\infty} := \inf_{A \subseteq \Sigma, \mu(A)=0} ...
0
votes
1answer
12 views

a bounded function is converge in measure, then its limit is also converge

If a series function, ${f_n} \rightarrow f $ in measure $\mu$, and $|f_n| \leq M$, how to show that $|f| \leq M$? My instructor gave a hint as follows, but I do not believe the first inequality ...
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0answers
37 views

Proving that any measure is the sum of a semi-finite measure and a measure which takes either 0 or infinity.

I need help proving that for a measure space (X, A, u) that u can be written as the sum of a semi-finite measure and a measure that takes on the values 0 and infinity. The second measure, u_i I ...
0
votes
1answer
45 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
0
votes
1answer
18 views

Measure of sum of sets of “Cauchy” sequence bounded?

Let $\{A_n\}_n$ be a sequence of sets of a $\delta$-ring $\mathfrak{M}$ of measurable sets with finite Lebesgue measure. Let us suppose that $$\forall\varepsilon>0\quad\exists ...