Ergodic theory lecture 6 continuous transformations. Probability measure on metric spaces article pdf available in journal of the american statistical association 63323 september 1968 with 1,065 reads how we measure reads. The topology of the space of probability measures on a metric. An open set in a metric space may be the union of many such sets. In this note we talk about the total variation metric and the narrow topology also called the weak topology on the set of borel probability measures on a metrizable topological space. The construction has been used to prove the existence of haar measures on compact groups see for instance page 3 of assaf naors notes.
This note investigates which of the topological properties oft. Polish spaces are so named because they were first extensively studied by polish topologists and logicians sierpinski. In the next two chapters, we will look at two important special cases of metric spaces, namely normed linear spaces and inner product spaces. The metric is usually called the gromovhausdorffprokhorov metric in the literature since it uses the. The boundary points are compact metric spaces x,d with. Structural aspects in the theory of probability series. A set p of radon probability measures on a locally compact metric space x,mis dobrushintight if. Parthasarathy, probability measures on metric spaces. Two such spaces are equivalent if they are isometric as metric. Probability measures and milyutin maps between metric spaces. A compact metric measure space is a compact metric space equipped with probability measure that has full support. Pdf probability measure on metric spaces researchgate. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. We prove that this kantorovich monad arises from a colimit construction on nite powerlike constructions, which formalizes the intuition that probability measures are limits of nite samples.
A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a. Several authors rediscovered these metric and norm later see short history in 16. Probability measures on metric spaces sciencedirect. Topological dynamics of transformations induced on the. It is stronger then usual continuity at every point because here depends only on the and not on the point nonexample. The narrow topology on the set of borel probability.
On a metric of the space of idempotent probability measures. Of course, if x, d is a compact metric space, then every. We then provide several examples of such structures, including the moyal plane, as well as compact quantum metric spaces and locally compact metric spaces. First, we work in metric spaces that are not necessarily separable. The next subsection provides more discussion on the matter. Probability measures on metric spaces prakash panangaden 3rd october 2019 these notes are heavily based on the book, \ probability measures on metric spaces by k. A metric space is complete if every cauchy sequence converges. Next, he describes arithmetic properties of probability measures on metric groups and locally compact abelian groups.
These notes are heavily based on the book, probability measures on metric. We then have the following fundamental theorem characterizing compact metric spaces. Let x be a complete separable metric space then every measure on xis tight. Lett be a continuous transformation of a compact metric spacex. Contents 1 algebras and measurable spaces 2 2 generated. The case of particular interest is studied, in which the underlying space is itself a space of finite signed radon measures on a metric compact, and the corresponding support is the convex set of probability measures. In the paper one obtains the asymptotic behavior of the finitedimensional diameters and of the. Last time we saw that mx is a topological space endowed with the weak topology which captures weak convergence. This is generalized in the literature to define the distance of two compact measured metric spaces. Sharp asymptotic and finitesample rates of convergence of. Probability and mathematical statistics, a series of monographs and textbooks. Entropy characteristics of spaces of probability measures on.
In any case, in a metric space, we have the notion of distance between points. Also, the haar measure on any compact group satisfies o 2 grekas, 1992b, propositions 1. Borel cross sections in locally compact groups 24 chapter iiprobability measures in a metric space 1. Metrization of the gromovhausdorff prokhorov topology. As described in the preface to that book, the importance of metric spaces for probability theory was emphasized by the ground breaking paper pro56 by. The reason for talking about metrizable spaces rather than metric spaces. Probability measures on metric spaces universiteit leiden. Probability measures on metric spaces 1st edition elsevier. Curvature of metric spaces department of mathematics at.
Probability measures and milyutin maps between metric spaces vesko valov 1 department of computer science and mathematics, nipissing university, 100 college drive, po box 5002, north bay, on, p1b 8l7, canada received 5 january 2008 available online 7 june 2008 submitted by jose orihuela abstract. We will also be interested in the space of continuous rvalued functions cx. Parthasarathy probability measures on metric spaces pdf be faced with convergence of probability measures on x. An important feature of this enlarged version is the emphasis on algebraictopological aspects leading to a wider and deeper understanding of basic theorems such as those on the structure of continuous convolution semigroups and the. The wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine. Ais a family of sets in cindexed by some index set a,then a o c.
The borel subsets of a metric space probability measures in a metric space probability measures in a metric group probability measures in locally compact abelian groups the kolmogorov consistency theorem and conditional probability probability measures in a hilbert space probability measures ov c0, 1 and d0, 1. R, probability theory on the closed subspaces of a hilbert space, les probabilites sur les structures algebriques. The visible theorem labels often show the theorem numbers in the book. Every continuous function on a compact set is uniformly continuous. A characterization of the barycenters of radon probability measures supported on a closed convex subset of a given space is presented. Informally, 3 and 4 say, respectively, that cis closed under. Linearly rigid metric spaces and kantorovich type norms. A metric space x is compact if every open cover of x has a. In these, the distance function is defined by a norm.
Hilbert space embeddings and metrics on probability measures. The space of borel probability measures on m with the prohorov metric d. A brief introduction, journal of mathematical sciences 140, no. Topological dynamics of transformations induced on the space. A of open sets is called an open cover of x if every x. A note on gromovhausdorffprokhorov distance between locally. The narrow topology on the set of borel probability measures. Given 2 n, let sn be a nite set of points xj such that fbxjg covers x. Besides of the total variation distance which can be introduced regardless the structure of the underlying measurable space, there are other sorts of metric spaces of measures. Parthasarathy probability measures on metric spaces pdf. Polish spaces are so named because they were first extensively studied by polish topologists and logicianssierpinski, kuratowski, tarski and others.
A typical distance between probability measures is of the type d. Let x be a compact metric space equipped with the borel. Categorical probability and stochastic dominance in metric. Existence and uniqueness of the gradient ow of the entropy. We close this section by recalling both the markovkakutani xed point theorem and its corollary asserting the existence of stationary measures for continuous actions on compact hausdor spaces. Weak convergence of probability measures on metric spaces. Borel cross sections in locally compact groups 24 chapter ii probability measures in a metric space 1.
Here is the most important theorem about tightness. Here is another useful property of compact metric spaces, which will eventually be generalized even further, in e below. My coauthor sean li and i recently ran across the seemingly not wellknown fact that every compact metric space is endowed with a canonical probability measure. The visible theorem labels often show the theorem numbers in the book, labels involving pm refer to the other book by. R, probability measures on metric spaces, new york. We do not develop their theory in detail, and we leave the veri. C0x be the banach space of continuous functions on x that vanish at in nity, x1. Chapter ivprobability measures in locally compact abelian groups 1. Continuous transformations of compact metric spaces. Probability measures on metric spaces onno van gaans. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. It is easy to see that solving this problem is equivalent to testing h0. Does every compact metric space have a canonical probability.
The book is conceived as a text accompanying the traditional graduate courses on probability theory. T probability measures on metric spaces of nonpositive curvature. On the compacity of the space of probability measures. Noncompact locally compact quantum spaces with candi. Purchase probability measures on metric spaces 1st edition.
This book deals with complete separable metric groups, locally impact abelian groups, hilbert spaces, and the spaces of continuous functions. Compact sets in metric spaces uc davis mathematics. For certain aspects of the theory the linear structure of x is irrelevant and the theory of probability measures on metric spaces supplies some powerful tools. Litvinov, maslov dequantization, idempotent and tropical mathematics.
Weak convergence of probability measures these additional notes contain a short overview of the most important results on weak convergence of probability measures. A metric space is sequentially compact if every sequence has a convergent subsequence. Probability measures on metric spaces prakash panangaden 3rd october 2019 these notes are heavily based on the book, \probability measures on metric spaces by k. Graduate students, lecturers and researchers may use the book as a primer in the theory of probability measures on groups and related structures. The topology of the space of probability measures on a metric space prakash panangaden 23rd october 2019 1 introduction recall mx, the space of all borel probability measures on x. Heat kernels and analysis on manifolds, graphs, and metric. It follows in turn that with respect to a generic probability measure on the cone. If you dont, you can prove it directly by showing that kis limitpoint compact5, which for metric spaces is equivalent to compact. As a consequence, on every complete metric space, every kantorovich duality problem admits an optimal solution. Borel cross sections in compact metric spaces 22 5. After a general description of the basics of topology on the set of measures, he discusses regularity, tightness, and perfectness of measures, properties of sampling distributions, and metrizability and compactness theorems. Many more details and results as well as proofs can be found in the german lecture notes \wahrscheinlichkeitstheorie. Polish spaces, and it extends a construction due to van breugel for compact and for 1bounded complete metric spaces. Let x be a locally compact separable metric space and b its usual borel.
Let x be any set with a nite complete measure dx, y a compact metric space. In this paper we investigate algorithmic randomness on more general spaces than the cantor space, namely computable metric spaces. C bx the banach spaces of bounded continuous functions on x and x2. In the mathematical discipline of general topology, a polish space is a separable completely metrizable topological space. The semigroup of compact metric measure spaces and its. Let x be compact metric and let be a probability measure on x,bx. Xx like distality, transitivity, mixing property etc. Section 4 is dedicated to the proof of existence and uniqueness of the gradient ow of the entropy in the space of probability measures over a compact metric space. Instead, dobrushin 24 introduced a notion of metrically tights sets of probability mea. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex. T induces in a natural way a transformationt m on the spacem x of probability measures onx, and a transformationt k on the spacek x of closed subsets ofx. If xis a compact metric space, it has a countable dense subset. In chapter 3 we extend the kantorovich monad of chapter 2 to metric spaces equipped with a partial order. Quantum metric spaces and the gromovhausdorff propinquity 3 weak topology on the set of regular borel probability measures.