All Questions
Tagged with metric-spaces measure-theory
49 questions
2
votes
2
answers
142
views
Is completion of measures equivalent to completion of sigma algebras as metric spaces with respect to measures?
An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\mathbin\Delta J)=0$ for $E,J\in B$. Then the completion of ...
- 85
0
votes
0
answers
55
views
Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
- 291
2
votes
0
answers
116
views
For Polish $X,Y$, $L^p(X,Y)$ is separable
Let $X$ and $Y$ be Polish spaces. Equip $X$ with a Borel probability measure $\mu_X$ and $Y$ with a metric $d_Y$. We can define the $L^p$ space as follows:
Definition. Define
$\begin{align}L^p(X,Y) = \...
- 305
1
vote
1
answer
82
views
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
- 835
0
votes
0
answers
616
views
The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
- 305
2
votes
0
answers
95
views
Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
- 657
0
votes
0
answers
97
views
What is a metric for weak convergence of finite measures on a non compact, complete and separable metric space?
Consider the set of finite positive measures on a complete, separable, but not compact, metric space $S$, endowed with the topology under which a sequence of finite positive measures $\{\mu_n\}$ ...
- 1
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
1
vote
0
answers
30
views
Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
1
vote
0
answers
87
views
Is this metric on the space of $\mu$-measurable functions complete?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
- 657
1
vote
1
answer
170
views
Billingsley convergence of probability measures - inequality used in Theorem 2
On Page 8, Billingsley defines $f(x)=(1-\rho(x,F)/\epsilon)^{+}$ where $\rho(x,F)$ is the metric distance from the set $F$. He then states $|f(x)-f(y)|\leq \rho(x,y)/\epsilon$ and goes on to use this ...
- 19
2
votes
0
answers
48
views
The world of non-weak*-topologies on $\mathcal{P}(X)$
Let $X$ be a metrizable space and consider $\mathcal{P}(X)$, the set of all probability measures on $X$.
Typically, the weak*-topology is considered on $\mathcal{P}(X)$, which is a very natural ...
- 429
2
votes
0
answers
130
views
Regularity of $\sigma$-finite measure pushforwarded by completion
Let $(X, d)$ be a metric space. Let $\mu$ be a $\sigma$-finite measure defined on borel subsets of $X$. Let $i \colon X \to \hat{X}$ be an isometry on image, where $\hat{X}$ is a complete metric space ...
- 820
2
votes
0
answers
150
views
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
- 657
2
votes
0
answers
92
views
A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged
I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
- 835
2
votes
1
answer
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
- 657
0
votes
1
answer
216
views
Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\...
- 657
4
votes
2
answers
374
views
Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
0
votes
0
answers
65
views
Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
- 835
1
vote
0
answers
450
views
Portmanteau theorem for finite signed Borel measures
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$,
$\mathcal M_1(X)$ the ...
- 657
1
vote
1
answer
61
views
Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$, and
$\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$.
Then $\mathcal ...
- 835
4
votes
0
answers
492
views
Disintegration of measures: a confusion about an existence proof from a lecture note
I'm reading a proof of Theorem 2.25 below from this note. First, we recall a definition and a theorem, i.e.,
Theorem 2.25 (Disintegration). Let $\left(Z, d_Z\right)$ and $\left(X, d_X\right)$ be ...
- 657
1
vote
1
answer
125
views
How to find the point at minimal average distance of a given measure
Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given ...
- 10.6k
5
votes
1
answer
129
views
Are normal metric currents dense in the space of all metric currents?
It is classical that Euclidean normal currents are dense in the space of all currents.
This can be achieved through mollification.
What I want to know if this is still true for metric currents.
In ...
- 391
6
votes
1
answer
551
views
Relationship between doubling constant of a metric space and of a metric measure space
Let $(X,d,m)$ be a metric measure space. We say that it is doubling in the sense of metric spaces if for every:
$x\in X$ and every $r>0$ there exists some (metric) doubling constant $C_d\geq 0$ ...
- 195
0
votes
1
answer
189
views
Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
- 5,405
2
votes
0
answers
302
views
Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
3
votes
1
answer
226
views
Expected measure of a ball in a probability space with a metric
Assume we are given a probability space $(\mathbb{X}, \mathcal{X}, \mathbb Q)$ and a measurable distance function defined on it $d:\mathbb{X}\times \mathbb{X}\to \mathbb{R}^+\cup\{0\}$ that conforms ...
- 33
1
vote
0
answers
52
views
A local base for space of probability measures with Prohorov metric
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
- 111
0
votes
1
answer
164
views
separable support of Borel measure, with tau-additive measure and full support
I have a problem with Proposition 7.2.10 in Bogachev's Measure Theory Volume II book on page 77 (I have link to my drive with that book https://drive.google.com/file/d/...
- 137
6
votes
2
answers
1k
views
Is the separability of the space needed in the proof of the Prohorov's theorem?
The Section 5 of the book:
Billingsley, P., Convergence of Probability Measures, 1999,
studies Prohorov's theorem. A short reminder is given below.
Let $\Pi$ be a family of probability measures on ...
- 657
7
votes
0
answers
282
views
Composition of couplings as a pullback construction
A metric measure space $(X,d,\mu)$ consists of a metric space $(X,d)$ together with a Borel measure $\mu$. A coupling between metric measure spaces $(X_1,d_1,\mu_1)$ and $(X_2,d_2,\mu_2)$ consists of ...
- 554
1
vote
1
answer
160
views
Comparing $(((A^\varepsilon)')^\varepsilon)'$ and $int(A)$, where $A' := X\setminus A$ and $A^\varepsilon := \{x \in X \mid d(x,A) \le \varepsilon\}$
Disclaimer. This is follow up to the question https://math.stackexchange.com/q/3486130/168758.
Let $X=(X,d)$ be a Polish metric space equiped with the Borel $\sigma$-algebra and let $\mu$ be a ...
- 6,853
0
votes
1
answer
268
views
Topologies and Borel $\sigma$-fields on disjoint unions
Consider a set of functions $\mathcal{F}$ on $E$ where $E \subset\mathbb{R}^k$ - e.g. the class of $L_1$ functions on $[0,1]$ - and endow it with a suitable metric $d$ that makes it Polish.
Consider ...
- 195
9
votes
4
answers
4k
views
Is the space of Radon measures a Polish space or at least separable?
Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
- 657
2
votes
1
answer
261
views
Bounded ball measure on compact metric space
Fix $c>1$. Let $(X,d)$ be a separable compact metric space, does there necessarily exist a Borel probability measure $\nu$ on $(X,d)$ such that
$\operatorname{sup}_{x \in X,r>0}\frac{\nu(\...
- 5,405
2
votes
1
answer
1k
views
Proving that family of sets has non-empty intersection
Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already:
$S$ is set of measurable ...
- 105
4
votes
1
answer
482
views
(non) separability of the power set
Suppose that $(X,2^X)$ is equipped with a non-atomic probability measure $\mu$ (the existence of such spaces is consistent with ZFC). This induces the $L_1$ pseudometric $\Delta$ on $2^X$, via
$\...
- 6,541
4
votes
0
answers
2k
views
Does rate of convergence in probability come from a metric?
In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
- 63
15
votes
2
answers
2k
views
Where does the Lebesgue differentiation theorem fail?
The Lebesgue differentiation theorem says that for certain metric spaces $X$ (see below), any Borel measure $\mu$ that is finite on bounded sets and any $f: X \rightarrow \mathbb{R}$ locally $\mu$-...
- 1,368
3
votes
1
answer
390
views
Measures on complete metric spaces for which all meager sets are null
On a complete metric space the collection of meager and comeager sets form a $\sigma$-algebra. There is a 'natural' measure you can put on this $\sigma$-algebra where the measure of a meager set is 0 ...
- 12.5k
6
votes
1
answer
248
views
Joint measurability of metric
I am trying to understand in which metric spaces the metric is jointly measurable.
There exist a metric space $(X,d)$ for which the Borel $\sigma$-algebra, does not coincide with the product Borel $\...
- 193
16
votes
1
answer
632
views
Does there exist a ``continuous measure'' on a metric space?
Let $X$ be a separable complete metrizable space. Does there exist a complete metric $d$ and a Borel measure $\mu$ such that
(a)
$\mu(B_r(x))<\infty$ for every open ball $B_r(x)$ of radius $r>...
user1688
13
votes
1
answer
3k
views
Does this metric have an official name? Lévy metric? Ky Fan metric?
Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...
- 6,287
3
votes
3
answers
895
views
Compactness of sigma-algebra for the $L^1$ metrics
Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space?
...
- 5,721
6
votes
0
answers
969
views
What relates to measure spaces as topological spaces relate to metric spaces ?
Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...
- 529
5
votes
2
answers
642
views
Is the Hausdorff metric on sub-$\sigma$-fields separable?
Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
- 6,287
3
votes
0
answers
277
views
For METRIZABLE spaces, do the Banach classes and Baire classes coincide?
In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...
- 456
13
votes
5
answers
4k
views
Measure of the support of a Borel probability on a metric space
Does the support of a Borel probability measure always have full measure in a metric space?
I know this is true for separable metric spaces, and locally compact metric spaces. Is it true in general?
- 163