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In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. This note…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
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…
The concept of a $ C $*-algebra-valued metric space was introduced in 2014. It is a generalization of a metric space by replacing the set of real numbers by a $ C $*-algebra. In this paper, we show that $ C $*-algebra-valued metric spaces…
We describe the interface between measure theoretic probability and causal inference by constructing causal models on probability spaces within the potential outcomes framework. We find that measure theory provides a precise and instructive…
Causal spaces have recently been introduced as a measure-theoretic framework to encode the notion of causality. While it has some advantages over established frameworks, such as structural causal models, the theory is so far only developed…
Probability metrics constitute an important tool in probability theory and statistics \cite{DKS91}, \cite{R91}, \cite{Z83} as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with…
The main purpose of this paper is to study complex valued metric-like spaces as an extension of metric-like spaces, complex valued partial metric spaces, partial metric spaces, complex valued metric spaces and metric spaces. In this…
In the footsteps of the book \textit{Measure Theory and Integration By and For the Learner} of our series in Probability Theory and Statistics, we intended to devote a special volume of the very probabilistic aspects of the first cited…
A general theory of resource-bounded measurability and measure is developed. Starting from any feasible probability measure $\nu$ on the Cantor space $\C$ and any suitable complexity class $C \subseteq \C$, the theory identifies the subsets…
In the framework of the generalized measure theory the decomposable probabilistic-valued set functions are introduced with triangle functions $\tau$ in an appropriate probabilistic metric space as natural candidates for the "addition",…
This review summarizes the historical development of probability measures in asset pricing, from early mathematical finance and state price theory to risk-neutral valuation, martingale measures, forward measures, stochastic discount…
These lecture notes contain an introduction to some of the fundamental ideas and results in analysis and probability on infinite-dimensional spaces, mainly Gaussian measures on Banach spaces. They originated as the notes for a topics course…
We study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the Wasserstein metric $W_p$, for $1<p<\infty$. Our main result is that each of these generalized value functions is a type…
We introduce a category Prob of probability spaces whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving…
We introduced the concept of a metric value set (MVS) in an earlier paper \cite{GM} and developed the idea further in \cite{AS}. In this paper we study locally $M$-metrizable spaces and the products of $M$-metrizable spaces. Finally we…
The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion…
The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…
In this paper, we first generalize a value distribution result of Lahiri and Dewan [4] and as an application of this result we prove a normality criterion using partial sharing of small functions. Further, in sequel normality criteria of Hu…
The p-values are often implicitly used as a measure of evidence for the hypotheses of the tests. This practice has been analyzed with different approaches. It is generally accepted for the one-sided hypothesis problem, but it is often…