1
vote
0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
3
votes
0answers
21 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
1
vote
1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
2
votes
1answer
49 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
0
votes
1answer
38 views

Representative elements in the symmetric difference metric

The symmetric difference is a natural way to quantify the distance between measurable sets: $$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$ This is a pseudo-metric because there may be ...
3
votes
2answers
61 views

The completeness assumption in Prokhorov's theorem

Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented: Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values ...
1
vote
1answer
33 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, ...
0
votes
1answer
18 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 ...
0
votes
1answer
34 views

Proof strategy - Borel $\sigma-$fields

How does one go about proving the following: Every open set $A$ in the topological space $(\mathbb{R}^d,\|\cdot\|)$ (with the norm topology) is the union of all the open balls $B_\epsilon(q)$ whose ...
4
votes
1answer
79 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
2
votes
1answer
47 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
1
vote
1answer
34 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 ...
2
votes
1answer
48 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
3
votes
1answer
80 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
0
votes
1answer
38 views

Borel-set, open, measurable function.

I have a questions about Borel sets. Here is how they defined in my book: Now they say that, the set consits of open sets. But it must not nececarrily be all open sets on X? The reason this ...
1
vote
0answers
32 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
1
vote
2answers
35 views

Space of probability measures total bounded?

I want to consider a space of probability measures on some set $\Omega$. It's complete (am I right?). But I don't know whether it's total bounded. Actually, I want to prove that the space of ...
1
vote
0answers
24 views

Regularity of measures proof

A probability measure $\mathbb P$ on a metric space $(S,d)$ is closed regular if $$ \mathbb P(A) = \sup \{ \mathbb P(F) : F \subseteq A, F \text{ - closed} \} \text{*}$$ with $A\in ...
2
votes
2answers
55 views

Open sets are $\mu*$-measurable in a metric space with a given condition

I have a homework problem that I'm very stuck on. The problem statement is as follows: "Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then ...
1
vote
1answer
72 views

Existence of $1/i$-dense subsets

Let $(X, d)$ be a compact metric space and $m$ be a Borel measure on $X$. Assume that $\lbrace A_i\rbrace$ is a nested sequence of subsets: $\dots\subset A_i\subset\dots\subset A_2\subset A_1$ and ...
1
vote
2answers
81 views

Measure of big discontinuities

Let $D\subset\left[ 0,1\right] $ be a dense set, and $\mu$ Lebesgue measure on $\left[ 0,1\right] .$ Suppose $f:\left[ 0,1\right] \rightarrow\left[ 0,1\right] $ is continuous at each point in ...
0
votes
1answer
58 views

Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
2
votes
1answer
54 views

measurability of metric space valued functions

Let's say that we have a measure space $(X, \Sigma)$ and a metric space $(Y, d)$ with its Borel sigma algebra. If $f_n: X\rightarrow Y$ is an arbitrary sequence of measurable functions, then I ...
0
votes
1answer
148 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
4
votes
1answer
299 views

Existence of limit for convergence by measure for Cauchy-in-measure sequence+completeness of metric space?

Sorry if this is the wrong place to put it. But this question come from a graduate level textbook and seems pretty hard to me, so I hope this is a good place. Anyway, this come from the book Real ...
2
votes
0answers
105 views

Completeness of metric space induced by outer measure (similar to Nikodym metric)

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$. Define a relation $\sim$ on $P(X)$ by ...
2
votes
1answer
181 views

Proving completeness of Nikodym Metric

I'm trying to prove completeness directly of the metric given by $d(A, B) = \mu (A \triangle B)$ on a finite measure space $(X, M, \mu)$. Edit: I should make clear that I'm referring to completeness ...
0
votes
1answer
38 views

Reparametrization of an absolutely continuous curve

If $\alpha : [0,1] \rightarrow \mathbb{R^n} $ is $C^1$ and $\alpha'(t) \neq 0$ for all $t\in[0,1]$ then there always exists a reparametrization in which $\| \alpha'(s) \| = 1$. Is there an equivalent ...
2
votes
1answer
66 views

Removing isolated points to get a perfect set

The motivating question is the following: If $F$ is a closed subset of $\mathbb{R}^1$, can one find a perfect set $E\subset F$ such that $m(E)=m(F)$ (in Lebesgue measure)? Define $F_0=F$ and ...
1
vote
1answer
92 views

$\sigma$-algebra generated by open sets coincides with $\sigma$-ring generated by open sets.

Under the topic of Metric spaces in my measure theory book I came across this definition that says: "Denote by $B$ the $\sigma$-ring generated by the class of all the open sets of X. The sets of $B$ ...
1
vote
0answers
95 views

lebesgue measure is metric outer measure

This question is driving me crazy. I need to prove that Lebesgue measure is metric outer measure. Unfortunately, I get lost. All I have is because $m$ is Lebesgue measure, $m^*(A \cup B) < ...
3
votes
2answers
79 views

Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
2
votes
0answers
142 views

Bounded Lipschitz Metric on Space of Positive Measures

The bounded Lipschitz metric ($d_{BL}$) metrizes the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ $$d(\mu, \nu) = ...
3
votes
1answer
102 views

Non-trivial Hausdorff measures for general metric spaces

For a metric space $(X,d)$ and a dimension function $\varphi:[0,\infty)\to[0,\infty)$ we can define a metric outer measure $H_\varphi$ which is $\varphi$-Hausdorff measure. Since it is a metric outer ...
1
vote
1answer
171 views

metric and measure on the projective space

Let $RP^n$ be the $n$-dim real projective space. I have the following four questions. What is the so called standard metric on $RP^n$? More generally, consider a metric space $M$ with an equivalent ...
2
votes
0answers
70 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
4
votes
2answers
379 views

Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
3
votes
2answers
199 views

A question on Tight probability measures (regular measure)

This is somewhat a basic question, but I'm having difficulty proceeding with a certain part of the proof. I was reading Billingsley "Convergence of Probability Measures", and I encountered the ...
1
vote
3answers
86 views

A condition that balls have finite measure

Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties: $\forall y \in X$, $\forall r \geq 0$, ...
2
votes
0answers
86 views

Regarding nowhere dense subsets and their measure.

A while ago it was made clear that a nowhere dense subset $P \subset [0;1]$ whose Lebesgue measure $\mu(P) = \mu([0;1]) = 1$ doesn't exist. But is it possible in principle to define a nowhere dense ...
5
votes
1answer
148 views

Proving $\mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$

Can someone please help me show, why in a compact metric space $(X,d)$ we have have $$ \mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$$ and $$ \mu(A)=\sup\{\mu(K) \mid K\subseteq A, K \text{ ...
4
votes
1answer
435 views

Metrizability of weak convergence by the bounded Lipschitz metric

Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = ...
8
votes
2answers
713 views

Prokhorov metric vs. total variation norm

Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
3
votes
1answer
154 views

Radon measure on a metric space X

If $X$ is a metric space and $0<\mu(X)$, where $\mu$ is a radon measure and $\mu(\{x\})=0$. Can we always split $X = X_1 \sqcup X_2$ into two disjoint sets where $\mu(X_1) = a$, for any $0<a< ...
4
votes
1answer
89 views

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
8
votes
1answer
393 views

Algorithms for computing or numerically approximating the Prokhorov metric?

I am interested in the following practical question: Given two measures (say those of two parametric distributions), is there an algorithm for computing the Prokhorov metric between them? The general ...
1
vote
1answer
120 views

Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
1
vote
0answers
101 views

what is this called? “difference of the function is less than the function of the difference”

Given: a metric $d$ an aggregate function $f$ some sets (or multisets or random variables) $X$,$Y$ What do we call: $d(f(X),f(Y)) \leq f( [d(X_0,Y_0) \cdots d(X_n,Y_n)] )\ \forall\ ...
4
votes
1answer
705 views

About the Wasserstein “metric”

I've just encountered the Wasserstein metric, and it doesn't seem obvious to me why this is in fact a metric on the space of measures of a given metric space $X$. Except for non-negativity and ...
7
votes
1answer
284 views

Lipschitz continuity of an integral

Let $(E,d)$ be a metric space, $\mathscr E$ be its Borel $\sigma$-algebra and $\mu$ be a $\sigma$-finite measure on $(E,\mathscr E)$. Let the function $p:E\times E\to\mathbb R_+$ be non-negative and ...