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

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Expectation of Nonnegative Random Variable - Measurability

Recall the result that for a nonnegative random variable $X$ on $(\Omega, \mathcal{F}, P)$, $$ E[X] = \int_0^\infty (1 - F(x)) dx, $$ where $F$ is the cdf of $X$. In many of the proofs I've seen for ...
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0answers
9 views

Are these two statements involving null sets and $L^2$ Bochner functions equivalent?

Suppose I have two functions $f, g \in L^2(0,T;L^2(\Omega))$ where we have some bounded domain $\Omega$. Suppose that $$\text{for almost all $t$,}\quad f(t) \leq g(t) \quad\text{almost everywhere in ...
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42 views

If the right side of $\int f\ d\lambda = \int f\ d\mu − \int f\ d\nu$ exists, does the left one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
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2answers
25 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
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1answer
53 views

Is there measurable function defined on unmeasurable set?

In my textbook, Lebesgue measurable function is defined as for every finite $a$, the set $\{x\in E:f(x)>a\}$ is a measurable set of $R^n$. And it further states $E=\{x\in ...
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1answer
10 views

Characterization of the space of integrable functions stable under multiplication

Let $Ω = (Ω,Σ_Ω,μ)$ be a measure space and let $L(Ω)$ be the space of integrable functions on $Ω$. For $f ∈ L(Ω)$, set $L_f(Ω) = \{φ ∈ L(Ω);~f·φ ∈ L(Ω)\}$. Has the space of all integrable functions ...
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2answers
21 views

If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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1answer
32 views

$\sigma(f_i, i \in I)$ ($f_i:\Omega \to \mathbb{R}$) where $I$ is uncountable

$\sigma(f_i, i \in I)$ ($f_i:\Omega \to \mathbb{R}$) where $I$ is uncountable contains only sets that can be written as $\{(f_{i_1},f_{i_2},...) \in B\}$ where $B \in ...
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1answer
23 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
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1answer
25 views

How can I prove that $f$ and $g$ are measurable functions [on hold]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
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2answers
17 views

Sigma algebra on a Cartesian product

$\Omega_1$ and $\Omega_2$ are countable sets. With $\mathcal P(\cdot)$ we denote a power set of a set. We need to proof that: $$\mathcal P(\Omega_1)\otimes \mathcal P(\Omega_2)=\mathcal P(\Omega_1 ...
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0answers
42 views

measurable sets and open intervals

Let $A$ be a Lebesgue measurable set in $\mathbb {R} $ with a positive measure. Then, show that for any positive real number $r $, there is an open interval $I$ such that $\operatorname{m} (A\cap ...
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1answer
36 views

Density of measurable sets in $\mathbb{R} $

Let $A$ be a Lebesgue measurable set in $\mathbb {R} $. We can classify the points in $\mathbb{R}$ as 3 disjoint subsets: density 0 points $A_1$, density 1 points $A_2$, otherwise $A_3$. By the ...
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2answers
55 views

Borel set of $\mathbb R^n$ with $n > 1$

According to various sources, the Borel set over $\mathbb{R}^n$ can be defined in several equivalent ways: For instance, it can be defined as the smallest sigma-algebra containing every open set of ...
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2answers
60 views

If one side of $\int f\ d\lambda = \int f\ d\mu - \int f\ d\nu$ exists, does the other one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
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1answer
27 views

Radon measure and a non-L1 function

This is a part of the exercise 7.17 in Folland's Real Analysis: Suppose $\mu$ is a positive Radon measure on a locally compact Hausdorff space $X $ with $\mu (X)=\infty. $ Show that there exists ...
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1answer
41 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
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0answers
26 views

Efficient methods of covering a surface [on hold]

I am a fabricator. I run a machine that cuts material with a drill bit that run over a surface. I would like to know how to mathematically find the time it will take to cross a surface. An example ...
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2answers
45 views

Does there exist a subsequence whose intersection has measure greater than $1/2$?

I ran across the following problem on this review guide. It is problem 1.25, though I've changed the wording slightly. The measure is implicitly Lebesgue measure. Let $E_n$ be a sequence of ...
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0answers
57 views

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if ...
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0answers
25 views

Support of Radon measures

I am reading Folland's Real Analysis. The following is the exercise 7.2.b. Let $X$ be a locally compact Hausdorff space with a Radon measure $\mu$. Show $x\in\text{supp}(\mu)$ iff $\int f \text ...
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1answer
16 views

Difference in $\mathscr{L}^1(\mu)$ and $\mathscr{L}^1(\mu^\nu)$

Can someone give me an example that indicates the difference between $\mathscr{L}^1(\mu)$ and $\mathscr{L}^1(\mu^\nu)$, with $\mu^\nu$ indicating the measure's completion. We have seen that ...
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1answer
20 views

Question about proof extending measure to complete measure

I am looking through a proof in Folland, for Theorem 1.9, which states: Suppose that $(X, M, \mu)$ is a measure space. Let $N = \{N' \in M : \mu(N') = 0\}$ and $M' = \{E \cup F : E \in M' \text{ and ...
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0answers
15 views

Weak continuity of K-L divergence function

If $P_n$ and $Q_n$ are two pmf's of a discrete set (say $A$) with common support and $P_n \to P$ and $Q_n \to Q$ where the convergence is pointwise here (even weak would be fine here I guess), then $$ ...
5
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1answer
51 views

Proving inner measure equal to outer measure if a set is measurable

I'm doing the problem 19 in Real Analysis of Folland like below: Let $\mu^*$ be an outer measure on $X$ induced from a finite premeasure $\mu_0$. If $E \subset X$, define the inner measure of ...
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1answer
19 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
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45 views

If a measure $\mu$ is less than a measure $\nu$ on a generating $\pi$-system, can we conclude that $\mu \leq \nu$?

Let $\mu$, $\nu$ be finite measures on the non-degenerate compact interval $[a, b] \subseteq \mathbb{R}$ provided with the Borel $\sigma$-algebra. It is well-known that if $\mu(B) = \nu(B)$ for every ...
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1answer
38 views

Measure theory : Lebesgue outer measure. [on hold]

Let $E$ be a subset of $\mathbb{R}$ and $m^{*}(E)=0$. Prove that $m^{*}(E^{2})=0$, where $E^{2} = \{ x^{2} : x \in Ε \}$. Can you give me some ideas or hints? Thank you!
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28 views
+150

Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
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9 views

Convolution of measures on a measurable group is associative

I've come across a statement in Kallenberg's Foundations of Modern Probability which claims this and only tells me to use Fubini's theorem. I am not very familiar with this topic and the text doesn't ...
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1answer
35 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
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1answer
29 views

Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
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1answer
36 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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30 views

Does $L^p(L^1([0,1]))$ make sense?

I'm examining several function spaces like $L^p(X,\mu)$ where $X$ is a Banach space. Is it possible to take $X = L^1([0,1])$ and then look at $L^p(X)$? The problem I have is that I don't know ...
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32 views

How to separate self-defining values from sigma?

$$\sum_{k=1}^{m} \sum_{j=1}^{n} a_kx_j^{b_k+b_i} = \sum_{j=1}^{n} y_jx_j^{b_i}$$ What I need to do is solve $a$ for every $i$ given ($i$ is between 1 and $m$), so their result won't be composed of ...
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1answer
20 views

Prove that a complete field defines a partition of a set

Let $\Omega$ be arbitrary set. Let $Q$ be a partition of $\Omega$. I already proved that the collection of all unions of the cells in $Q$ is a complete field $\mathcal{F}$ (complete field is ...
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1answer
121 views

Total variation measure vs. total variation function

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on ...
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2answers
39 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
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2answers
37 views

Does the signed measure based on a Jordan decomposition of a function with bounded variation depend on the decomposition?

Let $g_1, g_2, h_1, h_2 : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous. Define $$ \begin{align} f_1 & := g_1 - h_1 \\ f_2 & := g_2 - h_2 \end{align} $$ and suppose ...
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0answers
15 views

Connection regularity in measure theory and approximation in premeasure

In the measure theory lecture, we defined a measure-theoretic content as follows: $ \mu: \mathscr{C} \rightarrow [0,\infty]$ with the property being additive on disjoint sets and that the empty set ...
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1answer
36 views

What is a linear functional on continuous functions on the real line not given by a measure?

What is a positive linear functional on continuous functions on the real line not given by integration against a measure? I know that the dual of $C_c(\mathbb R)$ is the set of Radon measures, ...
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2answers
71 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
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1answer
19 views

Measure on Product Set

Consider a finite sequence of $\sigma$-finite measure spaces $(\Omega_i, \mathcal{F}_i, \mu_i)$. Constructing the product measurable space $$ (\Omega_1 \times \cdots \times \Omega_n, \mathcal{F}_1 ...
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44 views

Dimension of a measure in terms of linear form on continuous functions

So this might be a bit of a weird question, but here goes. It is well-known (Riesz representation theorem) that the dual space of continuous functions on a compact $K$ identifies with the space of ...
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1answer
22 views

basic notions of measure theory: differences?

Could you help me differentiating the following notions of measure theory: law, probability, probability density, probability measure, probability distribution, distribution, distribution function. ...
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1answer
44 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
3
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1answer
34 views

Joint Distribution Implies Independence…?

Consider the measurable space $(\mathbb{R}, \mathcal{B})$ and a probability space $(\Omega, \mathcal{F}, P)$. Define a finite sequence of random variables $X_1,\ldots,X_n: \Omega \to \mathbb{R}$. ...
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1answer
13 views

Equivalence of definitions of Gaussian Measure

Wikipedia's article on Gaussian measures notes this as the definition of Gaussian measures: $\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 ...
7
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1answer
75 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
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0answers
16 views

Sunrise Lemma, left and right maximal functions

Would anyone be able to provide a proof of the following lemma: Where $| \cdot |$ is the Lebesque measure and $M_L$ and $M_R$ are the left and right maximal functions defined on $\mathbb{R}$ ...