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

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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
14 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
16 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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14 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 $$ ...
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13 views

About my proof for proving inner measure equal to outer measure

I'm doing the problem 19 in Real Analysis 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 $E$ to be ...
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1answer
17 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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1answer
30 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
34 views

Measure theory : Lebesgue outer measure.

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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13 views

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
32 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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19 views

A set which is not a G_delta nor an F_sigma [duplicate]

At first I thought of the Irrationals as a set satisfying these conditions but it does not.Is there any set with the above-mentioned properties?
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31 views

Is the set of irrational numbers not a $G_{\delta}$? [on hold]

Is the set of irrationals a $G_\delta$ set or not?
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24 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
35 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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28 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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0answers
30 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
79 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
34 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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14 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
34 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
66 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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43 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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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 ...
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1answer
33 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 ...
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72 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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12 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 $\mathbb{R}$ ...
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20 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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1answer
29 views

Lebesgue integral of a ratio of Lebesgue densities

I need a hint to solve the following problem: $P$ is a probability mass on $\mathcal B(\mathbb R)$ with a Lebesgue density $h$, $f$ is another Lebesgue density. I need to show that $\int ...
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1answer
22 views

How does the Lebesgue measure measure non-cartesian product sets?

Consider the measure space $(\mathbb{R} \times \mathbb{R}, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda)$ where $\lambda$ is the Lebesgue measure on $(\mathbb{R}, \mathcal{B})$. Since ...
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1answer
29 views

$E(f(|X_n|))$ property implies uniform integrability? [on hold]

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
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2answers
61 views

Is a limit of measure of a sequence of sets equal to measure of limit of the sequence of sets?

I'm sitting at the same question desk as this: Limit of the measure of the converging sequence of sets. Actually, I can't prove it neither. PA6OTA gave a hint to show there is subsequence $A_{n_k}$ ...
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1answer
22 views

Normally Distributed = Absolute Continuity?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. A random variable $X: \Omega \to \mathbb{R}$ is said to have the standard normal distribution if it has the density $f:\mathbb{R} \to ...
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35 views

Product Topology and Borel-$\sigma$-algebra

Let $S=\left\{1,2,...,n\right\}$ be equipped with discrete topology and let $X=S^{\mathbb{Z}}$. Then the so-called cylinder sets $$ [s_0,s_1,...,s_m]_n:=\left\{x\in X: \forall 0\leqslant i\leqslant ...
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1answer
43 views

Union of a null set and a non-measurable set

Suppose $S$ is a non-measurable set (wrt the Lebesgue measure) and $N$ has measure 0. What can be said about $S\cup N$? Is it also non-measurable?
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67 views

What is the difference(s) between $(a,\infty)$ and $(a,\infty]$?

I am studying H. L. Royden's Real Analysis which includes some introduction to Measure Theory; and I encountered $(a,\infty]$ instead of $(a,\infty)$ for the first time! What is the difference(s) ...
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1answer
40 views

A problem on measure restriction

Definition of measurable space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of measure: Let $(\Omega, F)$ be a ...
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109 views

Find measure such that…

I've a very concrete problem I can't solve. Consider the following function $k: [0,1]^2 \to \mathbb{R}:$ $$ k(x,y)=\begin{cases} 1 &\text{if } y > x \\ -1 &\text{if } x- \frac{1}{2} < ...
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1answer
21 views

Proof of “triangle inequality” for outer measures: $\left|\mu^*(A) - \mu^*(B)\right| \le \mu^*(A \Delta B)$

We use the following notation. $X$ is a set, $\Omega_0$ is a ring of subsets of $X$, $\mu_0$ is a premeasure on $\Omega_0$, and $\mu^*$ is the associated outer measure, defined for subsets $A$ of $X$ ...
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19 views

Product Sigma Algebra, Explicitly

Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be measurable spaces and define the product sigma algebra on $\Omega_1 \times \Omega_2$ as $$ \mathcal{F}_1 \otimes \mathcal{F}_2 = ...
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5answers
125 views

What we get if we add 1/2 infinite times [on hold]

I want to know if this is correct We have this sums: $$S1=1-1+1-1+1-1+1-1+1-1...=\frac12$$ $$S2=1-2+3-4+5-6+7-8...=\frac14$$ $$S3=1+2+3+4+5+6+7+8...=-\frac{1}{12}$$ If we take ...
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1answer
25 views

If ${f_n}$ converges to $f$ in $L_p$ sense and to $f'$ point-wisely, does it mean $f=f' a.e.$?

The question came into my mind when I read a theorem from Kubrusly's "Measure Theory: a First Course", saying that if $f_n\rightarrow f'$ uniformly and $f_n\rightarrow f''$ in $L_p$ sense, then ...
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30 views

Why is this definition of the mesaure $\mu$ not properly?

Consider the measurable space $$ (\left\{a,b,c\right\}^{\mathbb{Z}},\mathfrak{A}), $$ where $\mathfrak{A}$ is the product topology on $\left\{a,b,c\right\}^{\mathbb{Z}}$ which is generated by the ...
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1answer
38 views

What is Skorohod's Represntation Theorem Saying?

From Wikipedia: Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. ...