Suppose that $(X,𝒜)$ is a measurable space and $Y$ is a metrizable, Souslin space. Let ${𝒜}^u$ denote the universal completion of $𝒜$. For $x\in X$, let $\underline{f}(x,·)$ be the lower semicontinuous hull of $f(x,·)$. If $f:X\times Y\to \overline{ℝ}$ is $({𝒜}^u\otimes ℬ\left(Y\right),ℬ\left(\overline{ℝ}\right))$-measurable, then $\underline{f}$ is $({𝒜}^u\otimes ℬ\left(Y\right),ℬ\left(\overline{ℝ}\right))$-measurable.

Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space. We confirm...

In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order $r$ and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.