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

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1answer
46 views

Measure $\mu_f$ for function $f(x)=x^2$

Let $X=[0,1]$ and $\mu$ the Lebesgue measure. Describe the measure $\mu_f$ for the function $f(x)=x^2$. By definition of $\mu_f$: For any Borel subset $A\subseteq R$, we have ...
4
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1answer
98 views

Finding simple functions to bound $\frac{xy}{(x^2+y^2)^2}$

I want to show that $$ \int\limits_{[0,1]\times[0,1]}\frac{xy}{(x^2+y^2)^2}\,d(\mu\times\mu). $$ equals $\infty$, where $\mu$ is the Lebesgue measure. I've tried to find simple functions that give ...
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1answer
56 views

Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
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1answer
341 views

Lebesgue/Jordan null set

Can't understand, what is it about.. what is difference between Lebesque and Jordan null set? As far as I understand, it is some set, which contains nothing, but so what, what is the joy?
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2answers
65 views

Lebesgue integral in two dimensions over fraction

Let $X=[-1,1],Y=[0,1]$, and $$f(x,y)=\dfrac{xy}{(x^2+y^2)^2}$$ for $x\in X,y\in Y$. Let $\mu$ be the Lebesgue measure. Does the following integral exist: $$\int_{X\times Y} ...
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1answer
43 views

Definition of the outer measure

Let $X$ be a set. By definition, for every sequence of sets (disjoint or not), an outer measure $\theta:\mathcal{P}X\rightarrow [0,+\infty]$ is a monotic, countably subadditive (hence subadditive) ...
4
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1answer
47 views

continuity of a map on $M(\mathbb{R}^n)$

Let $M:=M(\mathbb{R}^n)$ be the space of probability measures on $\mathbb{R}^n$ with respect to the Borel $\sigma$-algebra. Let $K\subset M$ be a compact convex subset. $K$ carries a natural ...
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1answer
385 views

$\sigma$-algebras and product topology

What can be said about $\sigma(T_1 \otimes T_2)$ and $\sigma(T_1) \otimes \sigma(T_2)$, when $T_i$ are topologies that aren't necessary second countable, and $\otimes$ denotes, at the left, the ...
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1answer
56 views

Measure of a set

Let $A \subset R$ such that for all open interval I, $m^* (A \cap I) < 1/2 L(I)$, where L is the length of a interval and $m^*$ is measure, prove that $m^*(A)=0$. I appreciate any hint to solve ...
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1answer
62 views

Moving boundaries for Ornstein-Uhlenbeck processes

Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes ...
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1answer
57 views

A problem of maximization in Banach spaces

Let $X$ be a Banach space and $K\subset X$ compact. Let $C(K)$ be the set of continuous function in $K$ and $\mu\in (C(K))^\star$ a non-negative measure. Assume that $f:K\to X^\star$ is a continuous ...
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1answer
56 views

Sigma-Algebra Created by Intervals

Let $\{x_i \}$ be a finite collection of numbers with $x_0=0< x_1< \ldots < x_n < x_{n+ 1}= 1$. Let $F$ be a sigma-algebra generated by the intervals $[x_0,x_1),[x_1,x_2),\ldots ,[x_n,1]$. ...
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3answers
785 views

What is the relation between weak convergence of measures and weak convergence from functional analysis

To keep things simple, we assume $X$ to be a polish space (think of $X$ as $\mathbb{R}^n$ for example). Let's denote with $P(X)$ the space of all Borel probability measure on $X$. We say ...
12
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1answer
102 views

Non-averaging set cannot have positive measure

Let $E \subset \mathbb{R}$ be non-averaging, that is, it does not contain any midpoints, so for any $a, b \in E$, $\frac{a + b}{2} \notin E$. It's claimed that as a consequence of the Lebesgue density ...
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1answer
183 views

Please help with definition for outer measure?

The following definition is taken from page 29/Folland 's Real analysis. In my understanding, this definition said : the outer measure of any set A belongs to X is the largest lower bound of the ...
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0answers
242 views

Sets $Q$ such that outer measure $\mu^*$ equals inner measure $\mu_*$ form an $\sigma$-algebra.

Suppose $\mathcal{A}$ is a $\sigma$-algebra and $\mu$ is a finite measure on $\mathcal{A}$. For all $Q \subseteq \Omega$, let: $\mu^*(Q) = \inf \{\sum_{n=1}^{\infty} \mu(A_n): \forall n, A_n \in ...
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3answers
542 views

Show $\int_{A}|f(x)|\mu(dx) < \epsilon$ Whenever $\mu(A) < \delta$

Hello all mathematicians!! Again, I am struggling with solving the exercises in Lebesgue Integral for preparing the quiz. At this moment, I and my friend are handling this problem, but both of us ...
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2answers
132 views

The set of points at which a real function is continuous is borel?

$f: \mathbb D(f) \subset \mathbb R \rightarrow \mathbb R$. I have to prove that $\mathcal C_f = \{x\in \mathbb R : \text{ f is continuous at } x\}$ is a Borel set. So I define $A_n=\{x∈\mathbb ...
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0answers
178 views

Measure theory, notation

This is from Avner Friedman's "Foundations of Modern Analysis": Let $\mu$ be a measure with domain $A$ and let $E_n$ ($n=1,2...$) be sets of $A$. Then $\mu (\underline{\mathrm{lim}}_{n \rightarrow ...
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1answer
557 views

Prove that set has zero Jordan content iff its closure has measure 0

Prove that set has zero Jordan content iff its closure has measure 0. I am having trouble with both directions , any tips would be great. THanks!
2
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2answers
60 views

Given a equivalence relation in a probability space we want to show events $A_1$,$A_2$ are equivalent if $P(A_1 \cap A_2) =\max\{P(A_1),P(A_2)\}$

Let $(\Omega,\beta, P)$ be a probability space. The events $A_1, A_2$ are equivalent in this probability space if $P(A_1 - A_2) \cup P(A_2 - A_1) = 0$. Show that if $A_1, A_2$ are equivalent then ...
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1answer
108 views

Measure spaces are proper subsets

I want to prove that $L^2$ is of the first category in $L^1$, thus I have to prove that $L^2$ is the countable union of nowhere dense subsets. The hint I get is: Take $g_n(x)=n$ for $0\leq ...
2
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1answer
244 views

Lyapunov Theorem for beginners

I study the subject of fair division (cake-cutting), and many papers contain a reference to a theorem by Lyapunov, which states that the range of any real-valued, non-atomic vector measure is compact ...
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votes
2answers
206 views

Why is the measure of the reals not zero?

I have followed the argument that rationals, being countable and ordered, can be covered by a convergent sequence of decreasing intervals. I am trying to understand why the same argument can’t be ...
2
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1answer
162 views

Has a probability density function a weak derivative

Assume I have a probability density function $\rho$ on $R$. (e.g $\rho \geq 0$ $\int \rho dx =1 $ $\rho \in L^1(R)$ ...). So $\rho$ is the density wrt the lebesgue measure. Now I try to understand if ...
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0answers
302 views

do all diffeomorphisms of $\mathbb{R}^n$ preserve measurable sets?

Let $f$ be the Cantor Lebesgue function and $g(x)=x+f(x)$. Then $g(x)$ is a homeomorphism from $[0,1]$ to $[0,2]$. But $g(x)$ maps a non-measurable set to a measurable set. Hence homeomorphism does ...
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0answers
46 views

when does the “Laplacian” of a $W^{1,2}$ function exist?

$\Omega $ is an open domain in $R^n$. Given a function $u \in W_{loc}^{1,2}\left( \Omega \right)$, we define a functional ${L_u}$ on $Li{p_0}\left( \Omega \right)$(Lipschitz functions with compact ...
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2answers
115 views

Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?

Let say I have a filtration $\mathcal F_i$ with $\mathcal F_1$ contained in $\mathcal F_2$, $\mathcal F_2$ contained in $\mathcal F_3$ and so on...$\mathcal F_n$. $X_i$ is a stochastic process, $X_i$ ...
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1answer
1k views

How to prove essential supremum is a norm

Let $f$ be a measure function on $X$. If there exists an $M>0$ such that: $\mu(\{t\in X: |f(t)|>M\})=0$, we say $f$ is essentially bounded. The infimum of all such $M$ is called the essential ...
3
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1answer
86 views

Let $(X,\mathcal{M},\mu)$ be a complete measure space. Show $\mathcal{J}:=\{A\in\mathcal{M}|\mu(A^{c})=0\}\cup\{\emptyset\}\}$ is a topology

Let $(X,\mathcal{M},\mu)$ be a complete measure space. Let $\mathcal{J}:=\{A\in\mathcal{M}|\mu(A^{c})=0\}\cup\{\emptyset\}$ 1) Prove that $\mathcal{J}$ is a topology on $X$ Any thoughts on how to ...
4
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1answer
88 views

A neat proof that the Lebesgue measure is rigid motion invariant.

I'm busy doing a small undergraduate maths project on the Banach Tarski paradox and I was hoping I could prove that a lebesgue measure is rigid motion invariant but I can't find an eloquent proof ...
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1answer
37 views

$p(X = c)=1$ then $E(X) = c$

Let $X$ be an aleatoric number. If $X \equiv c$ then $E(X) = c$. But if $p(X = c)=1$ how can I show, starting from the axioms of expectation or easy properties (e.g. Chebyshev inequality), that $E(X) ...
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5answers
562 views

What's the quickest way to see that the subset of a set of measure zero has measure zero?

It seems obvious but I am having a hard time explaining to myself why that is. Considering that in general, a subset of a measurable set need not be measurable. For instance, the Vitali subset of $[0, ...
3
votes
2answers
58 views

Measuring a subset of an interval with a basic property (the couch potato tale)

This is a basic illustration of a problem I need to then generalize. Below the problem and a little story that should clarify what I have in mind. I hope this is clear enough. Consider the $[0,1]$ ...
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2answers
144 views

Independence of sigma-algebras

Good day to everyone. While solving some problem of studying character I obtained some statement to prove, which is like following (this is my internal interest to prove it rigorously). Assume that ...
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0answers
54 views

Intuition behind Gaussian isoperimetric inequality

I was wondering whether or not there's an intuitive way of understanding the Gaussian isoperimetric inequality. I have been studying the Classical isoperimetric inequality and I finally understand it. ...
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1answer
80 views

Intersections of connected sets with piecewise smooth boundaries

Suppose you have two connected sets $S_1$ and $S_2$ in $\mathbb{R}^n$ with piecewise smooth boundaries, and whose intersection $S=S_1 \cap S_2$ has positive Lebesgue measure. Will the sets $S$, $S_1 ...
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2answers
67 views

Finding conditional variance

I know the marginal Variance of $\operatorname{Var}(Y) = E(Y^2)- (E(Y))^2$ and conditional variance of $\operatorname{Var}(Y|X)$ is $E((Y-E(Y|X))^2\mid X=x)$. I am trying to expand out the last ...
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1answer
94 views

Showing that $(X_n)$ is a submartingale if and only if $(-X_n)$ is a supermartingale

I was reading up on submartingales and supermartingales and saw this statement which I do not understand. A stochastic process $(X_n)_{n\geq 1}$ is a submartingale with respect to a filtration ...
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2answers
126 views

$m ( \{ x : f(x) > 0 \} ) = 0 \implies f = 0 $ almost everywhere

Suppose $f$ is non-negative measurable function. Put $E = \{ x : f(x) > 0 \} $. Say $m(E) = 0$. In other words, $E$ is null set. Then does it follow that $f = 0 $ almost everywhere ?
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1answer
46 views

Question about a problem solution.

Let us define $$ \int\limits_E f \, dm = \sup Y( E, f ) $$ where $Y(E,f) = \left\{ \int\limits_E \varphi \, dm: 0 \leq \varphi \leq f \right\}$ where $\varphi$ is a simple function. Here is the ...
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1answer
49 views

On finitely additive premeasures

Any interval of the form $(a, b]$ or $(a, \infty)$ is called an $h$-interval, where $a, b\in\mathbb{R}$ (see for example, Folland [1] page 33). Let $\mathcal{A}_h$ be the set consisting of all finite ...
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3answers
89 views

$f$ non-negative measurable function and $B \subseteq A$ both measurable $ \implies \int_{B} f \leq \int_{A} f $

Let us define $$ \int\limits_E f \, dm = \sup Y( E, f ) $$ where $Y(E,f) = \left\{ \int\limits_E \varphi \, dm: 0 \leq \varphi \leq f \right\}$ where $\varphi$ is a simple function. So, I'm trying to ...
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4answers
559 views

Is the outer measure of $A\cup B$ equal to the sum of their outer measures if $A\cap B=\varnothing$?

I understand that Lebesgue outer measure on $\mathbb R$ is not countably additive. But if there are two disjoint sets, does the outer measure of their union equal the sum of their outer measure? Can ...
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1answer
82 views

If the events $\{E_n\}$ satisfy a certain property show that $P(\cap_{i=1}^k E_i) > 0$

Let $\{E_n\}$ be events such that $\sum_{i=1}^kP(E_i) > k - 1$ then we want to show that $P(\cap_{i=1}^k E_i) > 0$. My approach for this problem is by contradiction. Suppose $P(\cap_{i=1}^k ...
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3answers
449 views

Question about Royden's proof that Lebesgue measure is countably additive.

In Royden's book, he gives the following book for the proof that Lebesgue measure is countable additive. I will just give a sketch of the proof. Let $\{E_k \}_{k=1}^{\infty}$ be a collection of ...
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1answer
76 views

Showing that the preimage of a continuous function on R is a σ-algebra

Let $f$ be a continuous function on $\mathbb{R}$. Define $\mathcal{A}=\left \{ E\subseteq \mathbb{R} : f^{-1}(E)\in \mathcal{B}(\mathbb{R})\right \}$. I want to show that $\mathcal{A}$ is a ...
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1answer
234 views

Co-countable measure on uncountable set

Let X be an uncountable set. And let $$\mathcal{A} =\{A\subset X: A \text{ is countable or } X-A \text{ is countable}\}.$$ Then it can be shown that $\mathcal{A}$ is a $\sigma$-algebra. Define ...
1
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1answer
60 views

Lebesgue measure of a lower-bounded set

Let $A \subset (0, 1)$ be a Lebesgue measurable set and let $k > 0$. Suppose that if $0 \leq a < b \leq 1$, then $\mu_L(A \cap (a, b)) \geq k\mu_{L}((a, b))$. Prove that $\mu_L(A) = 1.$ I ...
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1answer
430 views

What is the intuition behind Adapted Process

I am reading up on stochastic process and in particular adapted process. I know that if $X_t$ is $F_t$ measurable for each t, then it is an adapted process. But I do not understand the intuition ...