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

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2answers
180 views

Why Jordan measure is undefined?

$R^2$, $A=\{(x,\;y)\in R^2\colon 0\leqslant x\leqslant 1,\;0\leqslant y\leqslant 1\}$. Consider $X=A\cap Q^2$. Why for $X$, $m_e X=1,\;m_i X=0,\;m_e X\neq m_i X$? Especially i interested in why inner ...
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3answers
105 views

A measure space exercise.

Assume $f : X \to [0,\infty]$ I want to prove $$\sum_{x \in X} f(x)<\infty \Longrightarrow \{x \in X | f(x) >0\} \text { is a countable set}$$ Is it connected with finite property? Give me ...
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1answer
108 views

Measure integration problem

Assume $A_j,j\geq 1,j\in\Bbb N$ are measurable sets. Let $m \in N$, and let $E_m$ be the set defined as follows : $x \in E_m \Longleftrightarrow x$ is a member of at least $m$ of the sets $A_k$. I ...
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1answer
282 views

Riesz Representation Theorem from Rudin Step III

For those who have the Rudin: Real and complex analysis book at hand. I think I have an idea on step III of Riesz Rep. Theorem on page 43, but wasn't quite confident about my logic. In the proof of ...
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1answer
360 views

Example of Non Uniform Integrability

Consider a family of functions $\{f_n\}$, where $f_n: X \rightarrow \mathbb{R}_{\geq 0 }$, and a probability measure on $X$. Please provide an example in which all functions $f_n$ are integrable but ...
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1answer
227 views

Continuous, integrable function with integrable derivative in $\mathbb{R}$

While studying for my exams, I came across this question and I'm trying to think about an intelligent way to solve it (the context is Lebesgue integration): Let $f:\mathbb{R} \to \mathbb{R}$, a ...
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1answer
711 views

Intuition behind $\limsup$ and $\liminf$ for probabilities

I've come across these limits in Fatou's lemma, this got me massively confused. I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ...
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1answer
92 views

Reference to a proof on simple function

On the Real Analysis - Modern Techniques and Their Application (second edition) by Gerald Folland, page 47 i found this theorem: "Let $f$ a measurable function. Then exists a sequence $(\phi_n)_{n \in ...
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1answer
64 views

Proving some properties of a partition of a set by atoms

Let $\Omega$ be a finite set and $\mathcal{A}$ be a $\sigma$-Algebra. I want to show, that there exists one and only one single partition $\{\pi_1 ,\dots , \pi_n\}$ such that $\pi_i \in ...
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2answers
488 views

Sums and Products of Borel measurable functions

We say a function f is Borel measurable if for all $\alpha \in \mathbb{R}$ the set $\{x \in \mathbb{R} : f(x) > \alpha \}$ is Borel. If f, g are Borel measurable, then f+g is Borel measurable. ...
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1answer
270 views

Building a Bernoulli sequence where finite patterns repeat infinitely often

As a step in proving that the union of intervals $B_E \subset [0,1]$ (where $E$ is the set of infinite Bernoulli sequences, e.g. 0.0110...., such that some finite pattern repeats infinitely often) is ...
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1answer
217 views

Approximate limit

In some course notes on "evolution equations in probability spaces (etc)" I have the following definition: A point $z \in \mathbb R^n$ is called the approximate limit of $f$ at $x$ if for every ...
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2answers
289 views

convergence of average of iid random variables

Is the average of i.i.d. random variables with zero mean and finite variance convergent to 0 in probability or $L^2$? How do I show this? I am just beginning to learn convergence of random variables, ...
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2answers
26 views

Borel $\sigma$-algebra of subsets of [0,1]

Let sample space be [0,1]. let $\tau$ be the collections of all the open intervals on [0,1]. Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing $\tau$. Is the statement any ...
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1answer
40 views

Is the diagonal measurable

Suppose we have $X=Y=[0,1]$, $\lambda$ the Lebesgue measure on $[0,1]$, and $\nu$ the counting measure on $[0,1]$. Show that the diagonal $\Delta=\{(x,x):x\in X\}$ is $\lambda\times\nu$-measurable. I ...
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1answer
15 views

Green identity for measures with compact support

Let $\Omega\subset\mathbb{R}^N$ be a bounded, smooth domain. Assume that $\mu \in \mathcal{M}(\Omega)$ has compact support in $\Omega.$ Let $u\in W_0^{1,1}(\Omega)$ be a solution of $$ \left\{ ...
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1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
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1answer
40 views

Find $\liminf X_n$ where $X_n=1_{[n,n+1]}$?

My attempt: Suppose $\omega=n_0$. Then choose $N\geq n_0+1$.Threfore, $X_N(\omega)=0$. Therefore, $\inf_{k\geq N}X_k(\omega)=0$. Does it suffice to prove that $\liminf\limits_{n \rightarrow \infty} ...
0
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1answer
38 views

Given a pairwise disjoint collection, $\limsup A_n = \emptyset$?!

Let $(A_n)_{n=1}^{\infty}$ be a pairwise disjoint collection. $\limsup A_n = \emptyset$?! This is in relation to @StephenMontgomery-Smith's hint here. Trying prove $\sigma$-additivity from ...
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1answer
43 views

Using Kolmogorov's 0-1 law in proof of shift map being ergodic

Why should ${\cal E}_\theta$ be trivial?. I dont see how Kolmogorov's 0-1 law says that in this case we should take the 0 option. This is only mention of ${\cal E}_\theta$ in my notes I can find. ...
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2answers
61 views

Limit of an integral, as the measure of the region of integration approaches zero

Hi everyone: Let $f$ be a function defined on on open set $D$ of $\mathbb{R}^{N}$, $(n\geq1)$. Suppose that $(\Omega_{\varepsilon})$ is a family of measurable sets in $D$ such that ...
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1answer
65 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
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2answers
49 views

What does it mean to take a “vee product” of measures?

I'm reading a paper by Choski and Nadkarni called "The Maximal Spectral Type of a Rank-One Transformation". In it, they have a collection of measures $\mu_n$ on a space $(X, \mathcal{B})$, and then ...
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1answer
38 views

Question about almost sure convergence.

I am struggling a little to understand almost sure convergence in probability theory. I have taken some general measure theory and there we had abot convergence almost everywhere. Basically it was ...
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1answer
37 views

Random variables and integrals

Could someone please explain how this holds: $\displaystyle \int_{\mathbb{R^n}} f d\mu = \int_{\Omega}f(Y_n)d\mathbb{P}$ Does it use the following proposition? Furthermore how does ...
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1answer
40 views

Two questions on Fatou's Lemma

While reading the following paragraph from Real Analysis by Stein (I hope this does not breach any copyright; if so, I have to type it out), two questions occurred to me. In the proof of Fauto's ...
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1answer
28 views

Integration with respect to conditional measure?

Let $(X_n)$ be a Markov chain. For $i\in S$ my text defines $$N_i:=\sum_{n=0}^\infty \mathbf 1_{\{ X_n=i \}}$$ and then, as a part of a larger proof, claims that $$\mathbb E_i(N_i)=\sum_{n=0}^\infty ...
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1answer
20 views

A question about tail $\sigma $-algebras

How do I show formally that the event $\{w\colon\, \lim_{k\rightarrow\infty} X_k(w)$ exists $\}$ is in the tail $\sigma$-algebra of the sequence $X_1, X_2,\ldots$? Intuitively is quite obvious. The ...
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votes
1answer
31 views

Inclusion in the Vitali Theorem of Non-measurable Sets

In the following theorem, why does the inclusion $E \subseteq \bigcup_{\lambda \epsilon [-2b,2b]\bigcap \mathbb{Q}} (\lambda + C_E)$ hold? I'm not really sure how to describe this precisely, but I ...
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1answer
17 views

What events should be added for it to become sigma algebra?

Event A means that you will be late to the lecture. Event B means that the lecturer will be late. What events must be added for this system of events to become a $\sigma$ algebra? My thoughts: I think ...
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2answers
32 views

Regarding measurable functions

Let $(X,\mathcal A)$ be a measurable space and let $f:X\to \mathbb R$ and $g:X\to \mathbb R$ be mesurable functions. Let $G$ be an open subset of $\mathbb R^2$. We want to show that $\{x\in ...
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2answers
44 views

Iterating functions of expectations

We all know that $E[E[X]]=E[X]$. I was wondering, does it also hold that $E[g(E[X])]=E[g(X)]$ for "any" function $g$?
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1answer
18 views

Markov Processes: How to show $\int P(X_t\in B\mid X_0)dPX_0^{-1}=P(X_t\in B)$?

Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the ...
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3answers
45 views

A problem on Lebesgue dominated convergence theorem

I have the following 2 problems for homework, and I couldn't do the 1st one and need to check if my solution is correct for the 2nd one thanks 1) If $f$ is an integrable function on $ \Bbb R $ such ...
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1answer
44 views

What is the difference between $\sigma$-algebra, $\sigma$-ring, and field of sets?

I don't really understand the difference between these stuff. They look really similar. What is the difference between those? Which one do people use in measure theory, and probability related things ...
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2answers
32 views

function that doesn't belongs to $L_1$, but belongs to $L_p$ for $1<p\leq\infty$

Working on Bartle's book The Elements of Integration I found this exercise: Take $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, with $\mu$ as countable measure and define $f(n)=\dfrac{1}{n}$, prove that ...
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1answer
31 views

measure theory problem

I am stuck at part d) of this problem. Do you see how to show that f is measurable? I must show that $f^{-1}[-\infty,r)$ is measurable for all r. I am not sure how to do it. I would assume that it ...
0
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1answer
40 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
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1answer
38 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
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3answers
119 views

Hi I was thinking about a problem and have a question: [closed]

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
0
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1answer
25 views

If $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$

Suppose $H$ is a Hilbert space. Is it true that if $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h\in H$? Certainly if $x_n\to ...
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2answers
48 views

Is $C(X)$ dense in $L^p$?

Let $X$ be locally compact Hausdorff. Let $\mu$ be a complete measure on $X$. Is $C(X)$ dense in $L^p(\mu)$?
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1answer
41 views

$\int f d\mu<\infty$ iff $\sum_{n=0}^\infty 2^{-n} \mu(\{x \in X : f(x) \geq 2^{-n}\})< \infty$.

I have to prove this, but I really don't have any idea of how to start, I don't know which result or technique I could use. I would appreciate any hint or idea to prove this. Thank you. Let $(X, ...
0
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3answers
63 views

How to use Hölder's inequality to show $L_q$ is a subspace of $L_p$?

Suppose the measure of $X$ is finite. I want to show that $L_q(X)$ is a subspace of $L_p(X)$, where $1\le p\le q\le\infty$. I know I need to use Hölder's inequality, but I am not sure how do I use ...
0
votes
1answer
29 views

Everywhere continuous extension of a almost everywhere continuous function

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure. If $f$ is continuous outside a set $N$ of $\mu$-measure 0, does there exist an everywhere continuous $g$ such that $f = g$ on $X ...
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1answer
63 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
0
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1answer
74 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
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1answer
16 views

Is this true for any $L^p$ space?

Suppose $f\in L^p$ with $1\leq p<\infty$. Let $E_\alpha=\{x\mid|f(x)|>\alpha\}$. Then $$\lim_{\alpha\to\infty}\int_{E_\alpha}|f|^p d\mu=0$$ Any hints would be appreciated.
0
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1answer
25 views

For $(a,b)$, if $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) \cap - E)$ then $E$ in $\mathbb{R}$ is measurable

If $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) - E)$ for all open intervals $(a,b)$, then $E$ in $\mathbb{R}$ is measurable. How do I prove this? Totally stuck.
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1answer
24 views

Notation, abbreviation $a.s.$ measure theory

Could you tell me what $m - a. s.$ means in measure theory? Here $m$ is a measure. Thank you.