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

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Equality of measures

I have two measures $\mu$ and $\nu$ supported on compacts in $\mbox{int } \mathbb{R}^{n}_+$. Are there some sufficiently general classes of such measures for which $$ \int\limits_{\mathbb{R}^n_+} ...
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402 views

Why is this probability measure countably additive?

I am trying to prove that the following probability measure given in my book (Rohtagi) is countably additive. My analysis is a bit rusty so if someone can explain to me the properties of integrals ...
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507 views

Transition, marginal probability measures and probability measure on product space

Let $(\Omega_i, \mathcal{F}_i), i=1,2$ be measurable spaces. Their product measurable space is $(\Omega, \mathcal{F})$. Let $\mu_1$ be a probability measure on $(\Omega_1, \mathcal{F}_1)$, and let ...
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99 views

Showing that $\int \liminf_{n \to \infty} f_{n}(x) dx \le \liminf_{n \to \infty} \int f_{n}(x) dx$

I don't know why, but I find this problem counter intuitive to me. Prove that if $\{f_n\}$ is a sequence of measurable nonnegative functions on a measurable set $E$ and $f(x)=\liminf_{n \to \infty} ...
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502 views

prove that f is a characteristic function

$ ( X, \mu) $ is a complete measure space and $E_{n}$ are measurable sets such that $ \mu (E_{n}) < \infty$ for all $n$. Let $ \chi_{E_{n}} $ converge to $f$ in $ L_{1}$. Prove that $f$ is the ...
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220 views

If the absolute value of each function in a sequence has a Lebesgue integral that is bounded by a fixed number, will the limit function be integrable?

We have sequence of measurable functions $f_1, f_2, \dots$ such that $f_n \rightarrow f$ a.e. and $f$ measurable. If we know that $\int |f_n|d\mu < B$ for all $n$ ($B$ is fixed and finite). Also ...
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135 views

Can this function be extended to a measure?

I wanted to add a quick follow up question to one I asked earlier here. To summarize, there I let $\mathcal{R}$ be the set ring of subsets of $\mathbb{Q}$ consisting of finite unions of left-open, ...
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411 views

Optimal coupling of two measures

Let,$\mu $ and $\nu$ be two probability measures on $\Omega$ such that $|\Omega| < \infty$. let $(X,Y)$ be an optimal coupling. How to prove that the optimal coupling is not unique , by finding a ...
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542 views

General versions of (Second part of) Fundamental Theorem of Calculus

This question arose from a discussion here, and maybe it is of interest as such: We know the (second part of the) Fundamental Theorem of Calculus (FTC), but what conditions are really neccessary? ...
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139 views

Measure in the Riesz representation theorem for closed subsets

Is it true that in the Riesz representation theorem $\mu(F)=\sup\{\Lambda(f): f\in C_c, 0\leq f \leq 1, \operatorname{supp} f \subset F \}$ for every compact (or closed) subset $F$? (It is known ...
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225 views

Finiteness of conditional expectation if expectation is finite

I have $E(X) < \infty$. Under which conditions follows that $E(X|A)<\infty$ ? (A is an event of the form {$Y=y$} if it should matter) If I can use the formula $E(X|A)=\frac{E(X 1_A)}{P(A)}$ ...
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103 views

Absolute continuity of the kernel

Let $(X,d)$ be a metric space and $K:X\times \mathcal B(X)\to[0,1]$ is a stochastic kernel on $X$. We call this kernel absolute continuous with respect to a measure $\mu:\mathcal B(X)\to\mathbb ...
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517 views

Density of measure

Let $dx$ be the Lebesgue measure on $\mathbb R^d$. Let $u:\mathbb R^d\to{\mathbb R}\cup\{\infty\}$ is a non-negative and measurable function. The question is that, what are the conditions on $u$ so ...
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313 views

question on a notation in Big Rudin

I am wondering why Rudin used the following notation in his "Real and Complex Analysis". It is in Definition 8.7, as following. If $(X, \mathscr{S}, \mu)$ and $(Y, \mathscr{T}, \lambda)$ are two ...
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208 views

Signed measure of subsets of zero-measured sets?

Hey, I'm reading the proof of proposition 6.8 in Big Rudin, pg. 120. I'll just mention where my confusion lies: Suppose K is an arbitrary measure (complex, real, whatever). In the proof of property ...
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3answers
943 views

Question on the notion of a $\sigma$-algebra generated by a function

I've started learning about measure theory and I'm trying to get some intuitive grasp of the basic concepts. This is only succeeding partially so far. There is an exercise which I don't quite ...
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828 views

How to understand marginal distribution

Given a random vector $X: (\Omega, \mathbb{F}, P) \rightarrow (\prod_{i \in I} S_i, \prod_{i \in I} \mathbb{S}_i)$, is each component variable $X_i, \forall i \in I$ of the random ...
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174 views

Prove f is an integrable map

Let $f:[0,1] \rightarrow [0,\infty)$ be a measurable function such that: $\mu (\{x \in [0,1]: f(x) > t \}) \leq \frac{1}{t(ln(t))^{2}}$ holds for each $t>3$. Show $f$ is an integrable map.
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1k views

derivative of characteristic function

I came across an interesting problem but unable to see how to approach it. How do I use the dominated convergence theorem (LDCT), to show that first derivative of characteristic function of ...
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42 views

Measure concentrated at a point

What does "a finite random measure $\nu$ is concentrated at a point" mean? And in this case, what is equal to $\int_{\Omega} x d\nu$ ? Thank you.
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14 views

Continuous random variables and probability density function

OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called ...
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21 views

Properties of Stochastic Interval

I'm reading "Limit Thoerem for Stochastic Processes" and finding it hard to calculate the Stochastic interval.For example : In proposition 2.10,$T$ is a stopping time: If $A\in\mathcal F_0$,I need ...
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25 views

Problems in the integration limits to apply Fubini's theorem

If $f:(0,a)\rightarrow\mathbb{R}$ integrable function and $$g(x)=\int_{x}^a \dfrac{f(t)}{t}dt.$$ Then $g$ is integrable and $\int_{0}^a g(t)dt=\int_{0}^a f(t)dt$. I have to use Fubini's theorem but ...
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39 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
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21 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
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29 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
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15 views

If $X=\{0,1\}$, there exists an outer measure $\mu^*$ on $X$ such that $\mu^* \neq \mu^+$

Background Let $\mu^*$ be an outer measure on $X$ , $\mathcal{M}^*$ the $\sigma-$ algebra of all $\mu^*$ measurable sets, $\overline{\mu}=\mu^*\bigg|_{\mathcal{M}^*},$ and $\mu^+$ the outer measure ...
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40 views

Which are the measurable sets with respect to this construction of an outer dirac-type measure?

Let $x \in \Omega$ and $\mathcal{A} \subset\mathcal{P}(X)$ an at most countable set with $\emptyset, \Omega \in \mathcal{A}$. Let $$ \delta_x: \mathcal{A} \to [0,\infty], \quad A \mapsto ...
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41 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
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18 views

Sub Sigma-Algebra and measurability

If a random variable $X$ is measurable with respect to a sub $\sigma$-algebra (let's say $\beta_{1}$), such that $\beta_{1}$ $\subset$ $\beta$ , is $X$ -necessarily- measurable with respect to the ...
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8 views

Question about the positive variation of a signed measure

If $(X,\Sigma)$ is a measurable space and $v$ is a signed measure, I want to prove that $$v^+(E)=\sup\{v(F):F\subseteq E,F\in\Sigma\}$$ where $E\in\Sigma$. Let $X=P\cup N$ a Hanh descomposition for ...
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20 views

Non measurable subset of a positive measure set

I am self-studying measure theory and I have seen this theorem: If $A$ is a set of positive measure, then there exists a subset $D$ of $A$ that is non measurable. I am not sure how to prove it. I ...
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33 views

Proving basic properties of Hausdorff dimension and measure

I have two questions on basic properties of the Hausdorff measure and dimension which I've taken for granted for a while (I'm revisiting Falconer after about a year), but that I've never actually seen ...
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1answer
20 views

Examples of predictable processes

I am asked to prove that the following processes are predictable. I am used to looking at stochastic processes as sequences of random variables (by fixing time) or as a collection of paths (by fixing ...
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29 views

If $X:\Omega\to\Omega'$ and $f:\Omega'\to\Omega''$ are measurable and $f$ is injective, then $\sigma(X)=\sigma(f\circ X)$

Let $(\Omega,\mathcal{A})$, $(\Omega',\mathcal{A}')$ and $(\Omega'',\mathcal{A}'')$ be measurable spaces $X:\Omega\to\Omega'$ be $\mathcal{A}$-$\mathcal{A}'$-measurable $f:\Omega'\to\Omega''$ be ...
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37 views

Is a random variable constant iff it is trivial sigma-algebra-measurable?

I found a proof here for a measurable function (instead of probability theory's random variable) being constant if and only if the sigma-algebra generated by it is the trivia sigma-algebra, I think ...
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2answers
66 views

The length of a point and the interval

I think the length of a point is $0$, and since biunique corespondence between the points of [0, 1] and [0, 10], therefore I came to the conclusion that there is a same number of points between [0, 1] ...
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25 views

if $|f_n|<g \in L^1$, and $f_n \rightarrow f$ in measure, how do we know $\lim_{n\to \infty} \int f_n = \int f$

I know that a subsequence converges, but I am not even convinced that $\int f_n$ converges at all. They are all finite, but I am not certain how to bound them. I have considered working with $\int ...
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31 views

Obtaining the measure of a set from a limit of measures of open sets

Let $S$ be a Polish space and $\mathcal B$ the Borel $\sigma$-Algebra on $S$. Let $\mu$ be some finite measure on $(S,\mathcal B)$ and $A\in \mathcal B$ such that $\mu (A)>0$. Is the following line ...
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39 views

$X\in L^1$, then $\int_{|X|>n}XdP\to 0$ and $P(A_n)\to 0 \Rightarrow \int_{A_n}XdP\to 0$

I'm trying to prove the following: 1. Suppose $X\in L^1$, then $\int_{|X|>n}XdP\to 0$ Attempt: $$\int_{\Omega}|X|dP = \int_{|X|≤n}|X|dP+ \int_{|X|>n}|X|dP = M<\infty \space \forall n$$ ...
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115 views

Sigma algebra generated by a quadratic function

I have a little difficulty determining what $\sigma(F)$ looks like for $$F: [0,1] \ni x \mapsto 1- |2x^2-1| \in [0,1]$$ I know that $F(x)=F(y) \iff x = \sqrt{1-y^2}, \ \ x,y \in [0,1]$. $\sigma(F) = ...
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36 views

Hausdorff measure of a subset of $\mathbb R^3$

Let $f \in L^1_{\text{loc}}(\mathbb R^3)$. We define $A \subset \mathbb R^3$ as $$ A := \left\{ x \in \mathbb R^3 \, : \, \limsup_{r \to 0} \frac 1 r \int_{\mathbb B(x,r)} \vert f(y) \vert \, \mathrm ...
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9 views

Augmenting a filtration

I have a short question regarding the topic in the title. Let $(\mathcal{F}_t)$ be a filtration on some probability space. Let $(B_n)$ be a sequence of events such that$B_n \in ...
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19 views

Topology of weak convergence, linear functionals and probabilistic intuition

One very basic question regarding the topology of weak convergence. We know that given the following: $X$ metrizable topological space, $\mathcal{B} (X)$ Borel $\sigma$-algebra, $\Delta (X)$ ...
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30 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
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18 views

Countable partition of a probability space

I am trying to prove the following statement. Even though it seems almost obvious that it must be true, I am having trouble with making my arguments precise. Let $\{D_i: i\in \mathbb{N}\}$ be a ...
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18 views

Mean value theorem for sliding window of Lebesgue integral of integrable function

Take $f \in L^1(\mathbb{R})$ and define $g(x) = \int_x^{x+1} f(t) \, dt$. If $g(a) > 0$ and $g(b) < 0$, is it necessarily true that there is some $c \in [a,b]$ such that $g(c) = 0$? I feel as ...
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25 views

Coniditional expectation for bounded random variables

I need to show that for bounded random variables $X$ and $Y$ we have ${\rm{E}}[X{\rm{E}}(Y||\mathcal{F})]={\rm{E}}[Y{\rm{E}}(X||\mathcal{F})]$. The only property of conditional expectation I am aware ...
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22 views

Kolmogorov 0-1 law, Measure Theory

Suppose that $(X_1, X_2,...)$ is an independent sequence of random variables and $Y$ is measurable $\sigma(X_n,X_{n+1},....)$ for each $n$. Show that there is a constant a such that $P(Y = a) = 1$. I ...
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31 views

The geometry meaning of Riemann–Stieltjes integral [duplicate]

Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??