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

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A rather basic exercise in measure theory

Suppose $\mu_n$ is a sequence of probability measures on some compact space with the Borel sigma-algebra. Define a new function $\mu$ by the equation $\mu(A)=\sum_{n=1}^\infty 2^{-n}\mu_n(A)$ for ...
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488 views

Upper semi continuity of Lim sup

I have been asked to prove If $f$ is bounded, then $g(x)= \overline{\lim}_{y\to x} f(y)$ is upper semi continuous. This means somehow I have to show that for some $x_0$ $$\overline{\lim}_{x\to ...
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142 views

Integrable Functions in Probability

Let X be integrable and $A_n$ a sequence of subsets such that $ \lim_{n\to \infty} {P(A_n)} =0$. Show that $E X 1_{A_n} \to 0$.
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89 views

Comparing $\sigma$-algebras generated by 2 functions

I'm trying to solve the following exercise: Let $f,g:\mathbb R_+\rightarrow \mathbb R_+$ be given by $f(x) = \sum_{n=0}^\infty n\mathbb 1_{[2n,2n+2)}(x)$ and $g(x) = \sum_{n=0}^\infty\mathbb ...
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515 views

Easy application of the Dominated Convergence Theorem?

I am struggling with an application of the Dominated Convergence Theorem (DCT) which has cropped up a few times in various proofs I have been studying, in particular a proof about approximating ...
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92 views

Show that $P=\sum_{n=1}^{\infty}2^{-n}P_n$ is a probability measure

Given: $P=\sum_{n=1}^{\infty}2^{-n}P_n$ I am trying to show that it is a probability measure, and further, that: $\int_{\Omega}XdP=\sum_{n=1}^{\infty}2^{-n}\int_{\Omega}XdP_n$ for any non-negative ...
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369 views

About an integral over measurable sets

Let $(X, \Sigma, \mu)$ a measurable space and $f$ an integrable function. Show that if $(F_n)_{n\in\mathbb N}$ is a decreasing sequence of measurable sets and $F=\bigcap_{n} F_n$, then ...
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543 views

Borel measure of half-open and open intervals

the Borel set is the $\sigma$-ring generated by the open sets. One possible Borel measure on the real line is defined, for a closed interval, as: $$\mu([a,b])=b-a$$ But, from my understanding, ...
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142 views

Lebesgue measure and matrix notation problem

I have trouble with understanding following from my text book in Measures and Integral theory. Let T be an orthogonal $n\times n$ matrix. If $\lambda^{n}$ is the Lebesgue measure then we have: ...
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188 views

Dominated Convergence Theorem for $p=\infty$?

Is there some version of DCT for $p=\infty$. That is, is it true that if there is a sequence of measurable functions defined on an open set $\Omega$ in $\mathbb{R}^n$, $f_n$ converging pointwise to a ...
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508 views

Under what conditions is expectation value distributive?

We know that for two real numbers $a,b$ and two random variables $X,Y$ we have that $E(a X + b Y ) = a E(X) + b E(Y)$. Under what conditions is it also true that for any three random variables ...
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84 views

Is this true? Cardinality and subsets.

If you know that a given set O is countable. $\#O\leq \#\mathbb{N}$ Does this imply that the following statement holds? $\# O \leq \#\mathbb{R}$ I'm not sure, but I think it makes sense, ...
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889 views

Prove that Borel sigma field on R(d) is the smallest sigma-field that makes all continuous functions f:R(d)->R measurable.

Prove that Borel sigma-field on $\mathbb{R}^d$ is the smallest sigma-field that makes all continuous functions $f:\mathbb{R}^d \to \mathbb{R}$ measurable. How do I go about proving this? Thanks!
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81 views

Question about measurable sets $E_n$ such that $\lim_{n\rightarrow \infty}L^N(E_n) = 0$.

Let $E,E_n \subset \mathbb{R}^N$ be measurable such that $E_n \subset E, E$ is bounded domain, $E_{n+1} \subset E$ and $\lim_{n\rightarrow \infty}L^N(E_n) = 0$. Are there $K_n$ compact such that $E_n ...
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74 views

A natural question about convergence

Let $u,u_k \in C^{0}(K)$ where $K \subset \mathbb{R}^{n}$ is a compact set. Assume that $u_k \rightarrow u$ uniformly. Is these hypotheses sufficient to guarantee that \begin{equation} ...
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129 views

Can we extend this measure uniqueness theorem?

Let $\mu_1$ and $\mu_2$ finite measures on $\sigma$-algebra $\mathfrak B$ such that $\mu_1(X)=\mu_2(X)$, and $\mathcal A$ an intersection stable generator of $\mathfrak B$ such that ...
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162 views

Why is ergodicity of transformations only defined for measure-preserving transformations?

In ergodic theory, why does the defintion of an ergodic transformation $T$, why do I have to claim that it is measure-preserving? E.g. $T$ is ergodic if $\mathbb{P}(A) \in \{0,1\}$ for all $A$ ...
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109 views

sequence of open intervals

Let $ \displaystyle{ \{ (a_n , b_n) : n \in \mathbb N \} }$ a sequence of open intervals on $\mathbb R$ such that $ \displaystyle{[0,15] \subset \bigcup_{n=1}^{n_0} (a_n ,b_n) }$ for some $ n_0 \in ...
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109 views

lebesgue measure of $\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, 0 \leq y \leq 10, z \in \mathbb Z \} $

Find the lebesgue measure of the set: $$ \Bigl\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, \quad 0 \leq y \leq 10, \quad z \in \mathbb Z \Bigr\} $$ I think is a null set but for some reason I ...
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595 views

bounded measurable function is the uniform limit of a sequence of simple functions

Let $ f: \mathbb R \to \mathbb R $ a non-negative bounded measurable function. Prove that there exists a sequence of simple non-negative functions $ (f_n)_{n \in \mathbb N} $ such that $ f_n \to f$ ...
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328 views

Lebesgue integral vs area under a curve [duplicate]

Possible Duplicate: Lebesgue measure on Riemann integrable function in $\mathbb{R}^2$ Is the Lebesgue integral of a positive real function of a real variable equivalent to the Lebesgue ...
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293 views

Continuous Monotonic Increasing Function $F$ s.t. $F'=0$ a.e. $x$ on an Arbitrary Compact $K$ with Measure Zero and no Isolated Points

I'm trying to prove that for any $K\subset[0,1]$ such that K is compact and $m(K)=0$ with no isolated points, there exists a continuous, monotonic increasing function $f$ which maps $[0,1]$ onto ...
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50 views

Prove equality of integrals

Let $f(x,y) = \text{sgn}(x-y)e^{-|x-y|}$ (Where $\text{sgn}(t)$ is the sign of $t$) I want to prove the equation below. $$\int^\infty_0dx \int^\infty_0 f(x,y)dy = -\int^\infty_0 dy \int^\infty_0 ...
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54 views

Semicontinuity problem

If $A \subset \mathbb{R}^n$, is that claim true? $$\chi_A \text{ is LSC} \Longleftrightarrow A\text{ is open}$$ And then how can I prove it? ($\chi_A$ is characteristic function : if $x \in A$ than ...
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185 views

Limsup of Cauchy Random Variables

Suppose $X_n$ are independent Cauchy r.v.s. I'm trying to prove that $\limsup \log X_n/\log n = c$ almost surely for some constant $c$. I know that by Borel-Cantelli it suffices is prove that ...
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328 views

Distribution Functions of Measures and Countable Sets

Let $\mu$ be a continuous probability measure on $[0,1]$. Then, the function $g:[0,1] \to [0,1]$ defined by $g(x) = \mu([0,x])$ is called the distribution function of $\mu$. I have proved that $g$ is ...
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117 views

Compactness in $L^1$

Let $\mu(\cdot)$ be a probability measure in $X$. Consider a function $f: Z \times X \rightarrow \mathbb{R}_{\geq 0}$, with $Z \subset \mathbb{R}^n$ compact, such that: $\forall x \in X, \quad z ...
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174 views

Convergence in measure to zero with certain conditions.

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. For $c_n>0$ such that either $\lim_{n\to \infty}c_n=0$, or $c_n\geq c>0$ for all ...
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287 views

absolute measure unit of length

The measure unit of angles can be defined geometrically using compass and ruler, but the unit of length can not be defined absolutely so today we have different unit length used (feet, meter miles). ...
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441 views

Vitali Hahn Saks Theorem application

Let $(X,\mathcal M,\mu)$ be a measure space such that $ \mu(X)<\infty$. Suppose that $(f_n)$ is a sequence in $L^p(X)$ such that $|f_k|$ converges weakly to $|f|$ in $L^p(X)$. A solution to a ...
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62 views

Making a function out of several measurable functions

This is something I've been curious about. Suppose $(X,\mathcal{R})$ is some measurable space, and $X=\bigcup_n A_n$ where the $A_n$ are measurable, but not necessarily disjoint. On each of these ...
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528 views

Exception to monotone and dominated convergence theorem

If we consider the space $L^\infty ([0,1],\,dx)$, why is it that both monotone and dominated convergence fail? My first take on the problem was to consider the characteristic function $f_n$ of ...
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97 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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176 views

Removing a hypothesis when generalizing the Lebesgue measure

Let $f:\mathbb R\to\mathbb R$ be a continuous increasing function. Define the (generalized) length of (finite) semiopen intervals, $$ \begin{align} \lambda_f:&\{[a,b):a,b\in\mathbb R\,;\;a\leq ...
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112 views

$\int_{0}^{1} f(x,y)g(y)dy=0$ for a.e. $x\in [0,1]$ with $g\in C[0,1]$ implies $f=0$ a.e. on $[0,1]\times[0,1]$

I'm stuck on a measure theory problem and need some hints. Let $S=[0,1]\times[0,1]$ be the unit square in $\mathbb{R}^2$ and $f\in L^1(S)$. Suppose that for any $g$ continuous on $[0,1]$ we have ...
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88 views

measure theory convergence theorems for non-discrete indexing parameter

I need to prove that if $f \in L^{1}(\mathbb{R})$ then $\int |f(x+t)-f(x)|dx \to 0$ as $t \to 0$. I thought about approximating f with simple or continuous functions but then realized I couldn't apply ...
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164 views

Restriction of measure to rationals

Let $X = [0,1]$ and $\mathbb Q$ - the set of rational numbers. We take $X' = X\cap \mathbb Q$ and define a measure on it such that $\lambda(X'\cap (a,b)) = b-a$ for any $a,b\in X$. This ...
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236 views

The product of a measurable subset of $\mathbb{R}^n$ and a measurable subset of $\mathbb{R}^m$

I'm very new to undergraduate Lebesgue measure and am still having problems with it. I'm trying to prove the following: Let $A\subset\mathbb{R}^m$ and $B\subset\mathbb{R}^n$ and $A\times ...
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155 views

convergence of integrals

if $f$ is integrable, i wish to show $\frac{n}{2} \int_{-1/n}^{1/n} (f(x+y) - f(x))dy \to 0 $ as $ n \to \infty $ this looks to be very intuitive, but im having trouble proving it formally. any ...
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262 views

measurability of limit superior for functions

Suppose $(X,\mathcal {M})$ is a measurable space. For function $f(r,x): \mathbb R\times X\to\mathbb R$, suppose each r-section of it is $\mathcal M$-measurable. For constant $R\in\mathbb R$, do we ...
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14 views

Predictability of $\int^t_0 f(X_s)\,\mathrm ds$ where $X$ is a Lévy process

Let $X_t$ be a Lévy process and $f(x)$ some smooth function. Under what conditions is $$ Y_t = \int^t_0 f(X_s)\,\mathrm ds$$ predictable? Not sure how to investigate this. It is clearly adapted, so ...
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29 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
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24 views

Let $(X,\mathcal{F},\mu)$ be a measure space and let $g\in L^1((X,\mathcal{F},\mu))$.

Let $\phi:[0,1]\to\mathbb{R}$ defined by $$\displaystyle \phi(t)=\int_X \frac{t^3g}{1+t^2g^2}\ \mathsf d\mu$$ Show that $\operatorname{Im}(\phi)\subset\mathbb{R}$ and that $\phi$ is continuous. ...
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21 views

Lusin property (N) for functions of several variables

I just read in a paper by Martio and Zeimer$^1$ that smooth functions ($C^1$) of several real variables have the have the Lusin property (N). I have two questions. First, could someone give me a ...
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30 views

Exercise, show inequality(measure theory).

This exercise resembles what we do when we create the Lebesgue measure, but it is not quite the same. An interval can be any type: ...
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51 views

The outer measure on $X$ has a collection $M$ that is a $\sigma$-algebra

This is part of Caratheodory's Theorem taken by Real Analysis, Folland If $\mu^*$ is an outer measure on $X$, the collection $M$ of $\mu^*$-measurable sets is a $\sigma$-algebra. We first need to ...
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37 views

Is $f$ integrable if it is the limit of integrable functions with a uniform bound on their integrals?

Let $f_n$ is a sequence of measurable functions on a measure space $(X,\mathcal{B},m)$ converging pointwise to a function $f$. Suppose that $f_n$ is integrable for all $n$ and ...
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52 views

How to use induction to show that $\delta(\mathcal G_1), \ldots, \delta(\mathcal G_n) $ are independent?

I have proven that if the systems $\mathcal G$ and $\mathcal H$ are independent then so are the Dynkin systems $\delta(\mathcal G)$ and $\delta(\mathcal H)$. Now I'd like to generalize it to $n$ ...
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58 views

Is proving $m(E) < \epsilon, \forall \epsilon > 0$ equivalent to prove $m(E) = 0$?

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function ...
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23 views

Question on mutual singularity and absolute continuity of complex measures

I was presented these two somewhat similar questions from Folland's real analysis (second edition) dealing with complex measures and their mutual singularity and absolute continuity. They are 3.19 and ...