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We are dealing with definition of expectation of random elements taking values in metric space given by I. Molchanov and P. Teran in 2006. The approach presented by the authors is quite general and has some interesting properties. We present two kinds of new results:• conditions under which the metric space is isometric with some real Banach space;• conditions which ensure "random identification" property for random elements and almost sure convergence of asymptotic martingales.

In recent years, convergence results for multivalued functions have been developed and used in several areas of applied mathematics: mathematical economics, optimal control, mechanics, etc. The aim of this note is to give a criterion of almost sure convergence for multivalued asymptotic martingales (amarts). For every separable Banach space B the fact that every L¹-bounded B-valued martingale converges a.s. in norm to an integrable B-valued random variable (r.v.) is equivalent to the Radon-Nikodym...

Probability theory at the turn of the nineteenth and twentieth centuries was not treated as a branch of  mathematics, but as a part of physics. Therefore, the grounds of tasks and truthfulness rights probability often took place through experiments (which led to many erroneous statements). Following the formation and development of the theory of probability we observe some groundbreaking achievements that have enabled this area to stand on a new qualitative level. Such an achievement was undoubtedly...

A sequence of random elements $\{X_j,j\in J\}$ is called strongly tight if for an arbitrary $ϵ>0$ there exists a compact set $K$ such that $P\left({\bigcap }_{j\in J}[X_j\in K]\right)>1-ϵ$. For the Polish space valued sequences of random elements we show that almost sure convergence of $\left\{X_n\right\}$ as well as weak convergence of randomly indexed sequence $\left\{X_{\tau }\right\}$ assure strong tightness of $\{X_n,n\in \mathbb{N}\}$. For $L^1$ bounded Banach space valued asymptotic martingales strong tightness also turns out to the sufficient condition of convergence. A sequence of r.e. $\{X_n,n\in \mathbb{N}\}$ is said to converge essentially with...