Related papers: Probability monads as codensity monads
We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein…
The Giry monad on the category of measurable spaces sends a space to a space of all probability measures on it. There is also a finitely additive Giry monad in which probability measures are replaced by finitely additive probability…
We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…
Probability monads on categories of topological spaces are classical objects of study in the categorical approach to probability theory, with important applications in the semantics of probabilistic programming languages. We construct a…
This paper is a follow-up to the author's work "Topology of probability measure space, I" devoted to investigation of the functors $\hat P$ and $P_\tau$ of spaces of probability $\tau$-smooth and Radon measures. In this part, we study the…
We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the…
The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category…
The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean…
For a Tychonoff space $X$, the constructions $\hat P(X)$ and $P_\tau(X)$ of the spaces of probability Radon measures and probability $\tau$-smooth measures on $X$ are considered. It is proved that these constructions determine functors in…
The set of all idempotent probability measures (Maslov measures) on a compact Hausdorff space endowed with the weak* topology determines is functorial on the category $\comp$ of compact Hausdorff spaces. We prove that the obtained functor…
We generalize the classical Monge-Kantorovich duality--typically established for tight (Radon) probability measures--to separable Baire probability measures, which are strictly more general than tight measures on completely regular…
A characterization is presented of barycenters of the Radon probability measures supported on a closed convex subset of a given space. A case of particular interest is studied, where the underlying space is itself the space of finite signed…
Given a compact space $K$, we denote by $P(K)$ the space of all Radon probability measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. For nonmetrizable compacta $K$ even basic properties of $P(K)$ spaces…
We construct a compact Hausdorff space $K$ such that the space $P(K)$ of Radon probabiblity measures on $K$ considered with the weak$^*$ topology (induced from the space of continuous functions $C(K)$) is countably tight which is a…
In this paper we introduce a metrics on the space of idempotent probability measures on a given compactum, which extends the metrics on the compactum. It is proven the introduced metrics generates the pointwise convergence topology on the…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.
In recent times, there has been a growing interest in a structuralist understanding of probability, measure and integration theory. The present thesis contributes to this programme in three ways. First, we construct a commutative…
The Macdonald symmetric functions are used to define measures on the set of all partitions of all integers. Probabilistic algorithms are given for growing partitions according to these measures. The case of Hall-Littlewood polynomials is…
Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are…