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

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Approximation in non-compact interval

Suppose that $f$ is a continuous function defined on a interval $I\subseteq \Bbb R$. (a) If $I=[0,1]$ and $\epsilon \gt0$ is given show that there are finitely many constants $a_k$, $1\le k \le n$, ...
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48 views

Convergence of measures

Consider the measurable space $(\mathbb R,\mathscr B_{\mathbb R})$. Suppose that $\{\mu_n\}_{n=1}^{\infty}$ is a sequence of finite Borel measures and $\mu$ is a finite Borel measure on $\mathbb R$. ...
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93 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
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23 views

How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
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154 views

Prove Jensen’s inequality: $F(\frac{1}{μ(X)}\int f \,dμ) ≤ \frac{1}{μ(X)} \int F(f)\,dμ.$ [closed]

Let $(X,A,μ)$ be a finite measure space, and let $F : \mathbb{R} → \mathbb{R}$ be a $C^2$ function with second derivative $F'' > 0$. Let $f \in L_1(\mu)$ be real-valued. Prove Jensen’s inequality: ...
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48 views

$\sigma$-algebra generated by a set

I want to show that if $X$ is an uncountable set then $\mathcal{S}=\{\{x\}:x\in X\}$ generates the $\sigma$-algebra $\mathcal{A}=\{A\subset X: A$ is countable or $X\setminus A$ is uncountable$\}$. I ...
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49 views

Point with many prescribed Lebesgue densities

Let $E\subset\mathbb{R}$ be a measurable set and consider the function $f:(0,1]\to\mathbb{R}$ defined by $$f(x)=\frac{m(E\cap [-x,x])}{2x},$$ where $m$ denotes Lebesgue measure. Assume that $F\subset ...
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87 views

Are there any cases where $\mathbb E(|X|)<\infty$ and $\mathbb E(X)<\infty$ aren't equivalent?

I often see $\mathbb E(|X|)<\infty$ among the givens in a statement. That made me wonder: why not just demand $\mathbb E(X)<\infty$? In the light of the theorem Let $f$ be measurable. Then ...
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49 views

On the gist of $\sigma(X_1,\ldots, X_n)$

As far as I understand the reason we have $\sigma(X_1,\ldots, X_n)$ all over the probability theory is that it tells us what questions are answerable by $X_1,\ldots, X_n$. Say, we run an experiment ...
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284 views

Fat cantor set has positive lebesgue measure

This is probably a duplicate of some other question, but it's not immediately obvious which. The fat cantor set is constructed by removing smaller fractions of the center in each stage of the cantor ...
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1answer
58 views

Independent Events or Random Variables

First recall the following definition of independent random variables. Let $(X_t)_{t \in \mathcal T}$ be a set of random variables, where $\mathcal T$ is an arbitrary index set. Then $(X_t)$ is ...
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35 views

Necessity of a hypothesis in Scheffé's lemma

Scheffé's lemma states that if $f_n$ is a sequence of Lebesgue integrable functions (i.e. $f_n \in L_1$) that converges almost everywhere to another integrable function $f$, then $\int |f_n - f| \, ...
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1answer
43 views

Is the diameter of intersection of a set with a sphere of radius $r$ a measurable function of $r$?

I have to face to following problem: let $X$ be a separable metric space and $x_0 \in X$ fixed. Consider an open bounded set $A \subset X$. I want to know if the function $f: [0, \infty) \mapsto [0, ...
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25 views

Is the set of discontinuities of a distribution function on $\mathbb{R}^n$($n \geq 2$) bound to be countble?

By distribution function I mean function satisfying the following conditions: $\Delta_{a1,b1}\cdots\Delta_{a_n,b_n}F(x_1,\cdots,x_n) \geq 0$, where $\Delta_{a_i,b_i}$ is difference operator with ...
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43 views

inequalities concerning integration and measure

Let $f$ be a non-negative function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} f=1$. Let $p\in(0,1)$. Let $E$ be any measurable subset of $\mathbb{R}^n$. Prove that $$ \int _E f^p\leq ...
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44 views

Showing a set function to be measure

Let $(X,\mathcal A)$ be a measurable space and let $\mu:\mathcal A\to [0,+\infty]$ be a finitely additive set function such that $\mu(\emptyset)=0$. We want to prove that $\mu$ is a measure on ...
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51 views

$m(E) \geq 0$ instead of $m(E) = 0$?

There's this lemma in Real Analysis by Royden and Fitzpatrick that goes: Lemma 16: Let $E$ be a bounded measurable set of real numbers. Suppose there is a bounded countably infinite set of real ...
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63 views

A question on Lebesgue Measure

I am trying to show if $E_k \subset (0, 1)$, $(k=1,...,n)$, and $\sum_{1 \le k \le n}{\mu(E_k)} > n -1$, then $\mu(\bigcap_{1 \le k \le n}{E_k}) > 0$. Intuitively this seems like such an ...
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31 views

If $f_n\rightarrow f$ in $L(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, then $f_n\rightarrow f$ almost uniformly.

let $X=\mathbb{N}$, $\mathcal{X}=\mathcal{P}(\mathbb{N})$ and $\mu(E)=\sum_{n\in E}2^{-n}$ How to show: If $f_n\rightarrow f$ in $L(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, then $f_n\rightarrow f$ ...
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31 views

$f\in L_2(\mathbb{R})$. $\int^\infty_nf^2dx\rightarrow 0$ as $n\rightarrow \infty$?

Let $f\in L_2(\mathbb{R})$ How to show that: $\int^\infty_nf^2dx\rightarrow 0$ as $n\rightarrow \infty$ Next, I believe this is false for $f\in L_1(\mathbb{R})$. true?
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167 views

weak star convergence of signed measures vs convergence in Fortet-Mourier norm

There is a norm for signed measures given by $$||\mu||_{FM}=\sup_{f\in \mathrm{Lip}_1(X),|f(x)|\leq 1}\langle f,\mu\rangle.$$ This is usually called Fortet-Mourier norm (or more often metric, but it ...
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348 views

monotone class theorem, proof

I am having difficulty with this proof: It is the three sentences I have colored that is very difficult. Could someone please explain why they are true? red line: I understand "Since A is an ...
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1answer
40 views

$f_n\rightarrow f $ in measure if and only if $f_n\rightarrow f$ almost uniformly in specified measure

Let $\mu(E)=\sum_{n\in E}n^{-2}$. How to show that: $f_n\rightarrow f $ in measure if and only if $f_n\rightarrow f$ almost uniformly could you please help
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31 views

Prove that if (X, $\mathcal{A}$, $\mu$) is a measure space, $f$ is measureable $/iff$ $f^+$ and $f^-$ are measurable

Prove that if (X, $\mathcal{A}$, $\mu$) is a measure space, $f$ is measureable $\iff$ $f^+$ and $f^-$ are measurable$\mathcal{A}$ Where $f^-(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } ...
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1answer
104 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
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51 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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51 views

Expectation of the square of the minimum of iid positive random variables

Let $X_1, X_2$ be i.i.d., positive random variables with $E[X_i] < \infty$ (but $E[X_i^2]$ might be $\infty$). $Y := \min \lbrace X_1, X_2 \rbrace$. I want to show that $E[Y^2] < \infty$. The ...
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286 views

Riemann-Lebesgue lemma

How can I prove the following result? Let $([-1,1],M,m)$ a measure space, where $m$ is the Lebesgue measure in $[-1,1]$. If $f$ is Lebesgue integrable, then ...
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1answer
99 views

continuous function using the monotone convergence theorem

Let $f:\mathbb{R} \longrightarrow \mathbb{R}^+$ an integrable function. Defines $g(x)=\int_{-\infty}^{x}f(t) dt$. Show that $g$ is continuous using the monotone convergence theorem. I cannot find out ...
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98 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
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125 views

Calculating expectation conditioned on a sigma algebra

Let $\Omega=(0,\infty)$ and $\mathcal F=\mathcal{B}(\Omega)$. Let $\mathbb P$ be the probability measure corresponding to the exponential distribution with parameter $\lambda$. $X$ and $Y$ are two ...
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1answer
39 views

Hausdorff is a metric outer measure

I am new to measure& hausdorff measure, when looking at the proof of this property, I have a question : Given $E_1,E_2 \subset X,X$ is a metric space, we want to prove that if ...
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45 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
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79 views

A question about conditional expectations

Let $(\Omega, \mathcal{F}, P)$ be the underlying probability space. Let $\mathcal{G}$ be a sub-sigma-algebra of $\mathcal{F}$. Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a measurable function. ...
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1answer
25 views

Clarification on set notation for set of points where a given sequence converges.

Prove that, given a sequence of measurable functions $\{f_{n}\}$, the set of points at which $\{f_{n}\}$ converge is measurable. My solution is to first define $f(x) = \limsup_{n \to \infty} ...
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33 views

What is the name of this measure property?

if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ ...
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62 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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47 views

$f_n\rightarrow g$ in $L_1$ and $f_n\rightarrow h$ in $L_2$ .Then $g=h $almost everywhere

$f_n\rightarrow g$ converges in $L_1$ and $f_n\rightarrow h$ converges in $L_2$ how to show: $g=h$ almost everywhere Attempt: convergent in $L_1$ implies convergent in $L_2$. then by triangle ...
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41 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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109 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
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75 views

$E$ measurable if and only if $E \cap (a,b)$ is measurable for any interval $(a,b)$

We take the definition of measurability to be the following: $E \subseteq \mathbb{R}$ is measurable if for any $\varepsilon > 0$ there is an open set $G$ and a closed set $F$ such that $F \subseteq ...
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105 views

Where to find Geman 1995's proof on Changes of Numaraire?

Geman, H., El Karoui, N., Rochet, J.C. (1995) published paper "Changes of Numeraire, Changes of Probability Measures and Pricing of Options", on "Journal of Applied Probability " vol 32, pg 443-458. ...
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117 views

Measurable Subsets + Caratheodory Measurability

1.) What can go wrong if one assigns a measure to more subsets, especially to all subsets? (I would like to understand the subtleties behind) I imagine the first problem is to give the new subset ...
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87 views

References for a second course in probability theory

I need a probability book that treats all the arguments from the point of view of the measure theory and the Lebesgue integral. I've the basis of "naive" probability theory and of measure theory so I ...
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69 views

Lebesgue measure problem

Let $f$ be a non-negative measurable function on $\mathbb{R}$, and suppose that $\int f=0$. Prove that the set where $f \neq 0$ is a zero set. The hint says to let $E_n=\{f>1/n\}$ and then compare ...
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244 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
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55 views

In Egorov's theorem, remove the condition $\mu(E) < \infty$ and let the sequence be convergent in measure. The conclusion holds for subsequence

Let $(X,\mathscr{F},\mu)$ be a measure space, $E \in \mathscr{F}$, $\{f_n\}$ is a sequence of measurable functions on $E$, and the sequence converges to function $f$ in measure. Show that $\exists ...
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403 views

Continuous, strictly increasing function that maps a set of positive (lebesgue) measure onto a set of measure zero?

Is there a continuous, strictly increasing (real-valued) function on the interval $[0,1]$ that maps some set of positive (lebesgue) measure onto a set of measure zero? Should I play with cantor ...
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97 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be o-c?

Which are the conditions for a Lorentz space $L^{p,q}$ to be ord. continuous? ( A Banach function space is o-c $\equiv$ Increasing sequences of order-bounded positive functions converge in norm). ...
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52 views

Two measures on a same space

I have two measure space $(X, S, \mu_1)$ and $(X, S,\mu_2)$, where $S$ is the minimal $\sigma$-algebra containing sets $T = \{E_i\}_{i \in I}$. Suppose further that $T$ is closed under taking finite ...