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

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36 views

Any rule of thumb that says any reasonable function I can write down is measurable?

Question arose in the context of probability: If $Y$ is $F-$measurable and $h$ is some function, $h(Y)$ is $F-$measurable if $h$ is a measurable function (Doob-Dynkin). For example, let $f(x,y)$ be ...
2
votes
1answer
40 views

Fubini-Tonelli theorem and absolutely Lebesuge integrable functions

As far as I know, a measurable function is Lebesgue integrable if and only it is absolutely integrable. It is simply because the definition of the integrability requires each of the positive part and ...
2
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0answers
45 views

Union of uncountable family of $\sigma$-algebras

Suppose that $\{F_\alpha\}$ is an uncountable family of $\sigma$-algebras, and let $H=\bigcap_\alpha F_\alpha$. Is $H$ also a $\sigma$-algebra? Why or why not? I understand that the intersection of ...
2
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0answers
69 views

Will the generated sigma algebra have this property?

Lets say you have a measurable space $(\Omega, \mathcal{A})$. And a measurable function $X: (\Omega, \mathcal{A})\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$. We then know that for the sigma ...
2
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0answers
51 views

Prove that this function is in $L^\infty$ with $\lVert g\rVert_\infty \le C$.

My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise. Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in L^1(...
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62 views

How to understand $E(X\mid B)$ in the measure theory way

From undergraduate probability course, we learn $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ given $P(B)>0$. And we learn that if $(X,Y)$ has a joint density $f(x,y)$, we can calculate marginal density $...
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69 views

Simple exercise measure theory/$\sigma$-algebras

Is this right? Q: Find an infinite collection of subsets of $\mathbb{R}$ that contains $\mathbb{R}$, is closed under the formation of countable unions, and is closed under the formation of countable ...
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38 views

If $D_1\subseteq D_2$ then the generated sigma-algebras are such that $\sigma(D_1)\subseteq \sigma(D_2)$

Let $A$ be a set of subsets of $\Omega$. Then, the $\sigma$-algebra generated by $A$ is defined as $$\sigma(A):= \cap \{\mathcal{F}\; |\; \mathcal{F}\text{ is a }\sigma\text{-algebra and }A\subseteq \...
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42 views

Expectation of a step function and its extension to bounded above functions

Suppose that $$E_{\mu}[e^{-\alpha T}f(X)]=\int_{\chi}f(x)M_{\mu}(dx)$$ where $f$ is a measurable step function on Borel space $\chi$, $M_{\mu}$ is a nonnegative measure on $\chi$, $T$ (nonnegative)...
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0answers
40 views

Inclusion of $L^p$ and weak $L^p$ spaces

Let $0<p_0<p_1<\infty$, $0<\theta<1$, and $1/p_\theta=(1-\theta)/p_0+\theta/p_1$. Show that $$L^{p_\theta,\infty}(X)\subset L^{p_0}(X)+L^{p_1}(X).$$ Suppose that $f\in L^{p_\theta,\...
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1answer
32 views

Extending Borel Sigma Algebra “a little bit”

This is a follow-up of two previous questions discussed: Is every sigma-algebra generated by some random variable? Can every filtration be written as $\mathcal F^X$ for some process $X$ Consider ...
2
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0answers
50 views

Can every filtration be written as $\mathcal F^X$ for some process $X$

Given a stochastic process $\{X_t: t\in R^+\}$, which takes value in $R$, there is always a natural filtration $(\mathcal F^X_t)$ induced by $X_t$, i.e. $\mathcal F_t^X = \sigma(\{X_s^{-1}(A): s\le t, ...
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0answers
41 views

Defining a Sum of Random Variables

Given that a (real) random variable $X$ is a measurable map from a probability space $(\Omega,\mathcal{A},P)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, how do we define a sum of random variables $X+Y$...
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1answer
30 views

Representation of general Markov Chain

Let $(X,\mathcal{B})$ be a measurable space (state space) and let $T\colon X \times [0,1] \to X$ be measurable function that represents a dynamic in $X$ that has a parameter $u \in [0,1]$ (for a fix $...
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43 views

Probability space induced by a random variable

Consider three random variables $Y$, $X$, $Z$ all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $Y: \Omega \rightarrow \mathcal{Y} \subseteq \mathbb{R}$, $X: \...
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25 views

Change of variables in integration w.r.t Haar measure

Let $\Phi:[0,2\pi] \mapsto T$ be such that $\Phi(t)=(\cos t,\sin t)$, where $T$ is the unit circle and $\mu \left(S\right)=m(\Phi^{-1}\left(S\right))$ where $S$ is a Borel set of $T$ and $m$ is the ...
2
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1answer
31 views

Lebesgue integrability of the maximal function.

I'm practicing doing some questions on measure theory and I'm having great trouble with them. However, I've tried the following question and it seems quite easy hence I imagine I've probably made a ...
2
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0answers
34 views

Doob's $L^p$ inequality - Case $p = 1$

I have found in the wikipedia page following generalisation of Doob's so-called $L^p$ inequality, for general nonnegative submartinagles $X_s$: $$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + E[X_T\log^...
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39 views

Measure theory proof verification. A specific property of sets with positive measure and the Lebesgue measure.

I have a proof for the following question, could someone please verify it as i'm not confident whether its right. Let $E\subset \mathbb{R}$, such that $m(E)>0$, and $m(E^c)>0$ where $m$ ...
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0answers
17 views

Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation. Consider a measurable set $A\subset X$ with $0<\mu(A)<1$ For each $N$ define ...
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0answers
19 views

Summation property of Lebesgue outer measure. [closed]

Let $U$, $V\subset \mathbb R^N$ are such that $U\subset \{x_1\geq 0\}$ and $V\subset \{x_1<0\}$. I want to prove or disprove that $$ \mathcal L^\ast(U\cup V)=\mathcal L^\ast (U)+\mathcal L^\ast(V),...
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0answers
50 views

Showing Lebesgue's definition of measure implies measurability

19. 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 $\mu_*(E) = \mu_0(X) - \mu^*(E^c)$. Then $E$ is $\mu^*$-...
2
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0answers
41 views

Question on the Riesz -Markov Representation Theorem with composition of functions

I'm taking an introductory measure theory course as an economics student and I've quickly found that I'm not as prepared as I should be for this level of course. I have little experience in proof ...
2
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0answers
49 views

Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} \mathbb{...
2
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1answer
48 views

Is every measure finitely additive?

The answer is yes because for disjoint $A_i, i=1,2,...n$ $\mu(A_1\cup A_2 \cup ... \cup A_n) = \mu(A_1\cup A_2 \cup ... \cup A_n \cup \emptyset \cup \emptyset \cup ...) = \sum_{i=1}^{n} \mu(A_i) + \...
2
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0answers
68 views

Generalization of Tonelli's Theorem for Series

Let $A, B$ be sets and $x_{n,m}$ $n \in A, m \in B$ be a doubly infinite sequence of extended non-negative reals indexed by A and B. Show that $\sum_{(n,m) \in A \times B} (x_{n,m})$ = $\sum_{n \in ...
2
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1answer
31 views

Property of conditional expectation operator in $L^1$.

Let $\mathscr{G}\subset\mathscr{F}$ be two $\sigma$-algebras. It's easy to see that the conditional expectation operator $$E[\,\cdot \mid\mathscr{G}]\in \mathscr{L}(L^1(\mathscr{F}))$$ satisfies $\|E[\...
2
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0answers
43 views

Integration by parts for Dirac measure

We know that Dirac measure is defined by $$\delta_x(A)= \begin{cases} 1 &\text{if $x \in A$}\\ 0 &\text{if $x \notin A$}\\ \end{cases}$$ We know that $\int_a^b f(y) \, d\delta_x(y)=f(x),$ if ...
2
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0answers
35 views

Distribution of specific r.v. X & Properties of the Lebesgue Measure

Let $X(\omega)=a\omega+b$, in the $\left([0,1],\mathbf{B}(\mathbb{R}),\mu_{[0,1]}\right)$ probability space, where $\mu_{[0,1]}$ is the Lebesgue measure restricted to the interval $[0,1]$. How can I ...
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40 views

Weak convergence of measures time functions

Let us consider a sequence of measures $(\mu_n)_n$ which converges weakly to the measure $\mu$ on a metric space $X$. To simplify let us take $X=\mathbb{R}^n$ and $L$ be the Lebesgue measure. Now on $...
2
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1answer
32 views

Inclusions for certain types of measures

Let's use the following definitions: Definition. A measure $\mu: \mathcal P(X) \to [0, \infty]$ is what some authors call a outer measure, i.e. (1) $\mu(\emptyset) = 0$. (2) If $A, A_k \...
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116 views

Convergence a.e., and measurability

This is a question I find, but I think is very vague; maybe I shoudn't give an specific sequence of measurable functions and I need to give more conditions. The answer is correct but I'm not sure if ...
2
votes
2answers
65 views

Is $\sum Y_j$ a measurable $\sigma(X_1,…,X_n)$ function?

Let $X_n=\left(\sum^n_{i=1} Y_i\right)^2$, and I would like to know if $\sum^n_{i=1} Y_i$ is measurable in $\sigma(X_1,...,X_n)$. If not, why? I tend to think that it's not. But I can't think of a ...
2
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0answers
37 views

Meaure-theoretic induction: Why dyadic approximation?

In measure-theoretic induction proofs we always use the dyadic approximation of a non-negative measurable function $Y$ as $$Y_n = \sum_{k=0}^{n2^n-1} k/2^n 1\left(\frac{k}{2^n} \leq Y < \frac{k+1}{...
2
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57 views

Any illustration of a sigma algebra?

Let $X$ be a non-empty set and assume $\Gamma$ is in $2^X$ satisfying in $A\in \Gamma$ implies $A^c$ is also in $\Gamma$. We say $A$ is a $G^{\Gamma}_{\delta}$-set if there is a sequence $\{A_n\}$ in ...
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0answers
45 views

Caratheodory's method

The theorem is: Let X be a set and $\theta$ an outer measure on X. Set $\sum= \{E: E \subseteq X, \theta A=\theta(A \cap E)+ \theta(A \backslash\ E)$ for every $A \subseteq X\}$ What is the ...
2
votes
1answer
29 views

Convolution as a $L^1$ limit of translates.

I would like what convolution is, as a $L^1$ limit. Namely let $f,g\in L^1(\mathbb{R})$ (with some further conditions). Then what conditions on $f$ and $g$ ensure that $f\ast g$ is the $L^1$ limit for ...
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49 views

Disproving that a particular space is Banach

Let $E$ be a measurable set of finite measure and $1\leq p_1<p_2<\infty$. Consider the linear space $L^{p_2}(E)$ normed by $\|.\|_{p_1}$. Is this normed linear space a Banach space? My ...
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63 views

The set $\{x \in \mathbb{R} \mid m(f^{-1}(x))>0\}$ has measure zero.

The following question appeared on the Northwestern University Spring 2013 preliminary exam in analysis which is freely available online. Show that, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is ...
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74 views

Fatou's lemma. Case of convergence in measure

Fatou's lemma: Let $f_1, f_2, f_3, \cdots $ be a sequence of non-negative measurable functions on a measure space $(S,\Sigma,\mu)$. Define the function $f:S\to [0,\infty]$ a.e. pointwise limit by $$f(...
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0answers
62 views

Integral equation: averaging

Let $X, Y, Z$ be Borel spaces, for simplicity we can assume that they are $\Bbb R$. Consider an equation $$ \int_{X\times Y}f(x,y,z) \kappa(x,\mathrm dy)\mu(\mathrm dx) = \int_{X\times Y}f(x,y,z) \...
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0answers
39 views

Convergence in measure implies convergence in Frechet's distance

Let $(X,S,\mu)$ be a finite measure space and let's define: $$d(f,g):=\int \frac{|f-g|}{1+|f-g|}\,d\mu\;\;\forall f,g\in M(X,S)$$ I want to prove that if $f_n\ \xrightarrow[\mu]{} f\Rightarrow d(...
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0answers
38 views

How to use dominated convergence on $\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x) $?

I find it hard to find an appropriate dominating function for the integral $$I:=\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x), \ t > 0 $$ ...
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0answers
67 views

When can we restrict/condition a probability measure to a subset of zero measure?

If $\mu(C)>0$, $$\mu_C(A)=\frac{\mu(A\cap C)}{\mu(C)}$$ is well-defined. If $\mu(C)=0$ things get hairy. If $C=\{x\}$ (single point) then $\mu_C=\delta_x$. If $C=\bigcup_{i=1}^nx_i$ (a finite ...
2
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0answers
83 views

Continuous strictly increasing function is absolutely continuous iff set of infinite derivative maps to measure zero set

If $u:[a,b]\to\mathbb{R}$ is continuous and strictly increasing, prove that $u$ is absolutely continuous iff it maps $E:=\{x\in[a,b]:u'(x)=\infty\}$ into a set of measure 0. This question came from ...
2
votes
1answer
94 views

A continuous function defined by Lebesgue measure

Let $A,B\in\mathcal A_{\Bbb R}^*$ given with $\overline{\lambda}(A)<\infty$ and $\overline{\lambda}(B)<\infty$. Lets define $\; \overline{\lambda}_{A,B}:\Bbb R\to\Bbb R$ as follows: $$\overline{...
2
votes
1answer
55 views

Prove that the infimum is attained for outer measures

Let $(X, \Sigma, \mu)$ be a measure space. Define $$ \mu^*(S) = \inf\{\mu(U)|U \in \Sigma, S \subseteq U\}: \mathcal{P}(X) \to [0, \infty] $$ Theorem: $\forall S \subseteq X, \exists U \in \...
2
votes
1answer
31 views

Random variables on a probability space

I need to solve the following problem. Let $X$ be a random variable defined on the probability space $(\Omega,\mathcal{F},\textbf{P})$. Show that $X^{+} = max(0, X)$ and $X^{-} = max(0, -X)$ are ...
2
votes
1answer
29 views

Proof step in Rademacher's Theorem

In the proof of Rademacher's theorem, we assume that $f: \Bbb R^n \to \Bbb R$ is a Lipschitz function and $v \in \Bbb R^n$ is a vector with $\Vert v \Vert = 1$. Our aim is to show, that $$ \mathrm ...
2
votes
0answers
32 views

Ergodicity under measure-theoretic isomorphism

Suppose we have two measurable dynamical systems $(X_1,\mathcal{B}_1,\mu_1,T_1)$ and $(X_2,\mathcal{B}_1,\mu_2,T_2)$, with $\mu_i(X_i)=1,\ i=1,2$. Suppose they are measure-theoretically isomorphic (...