Continuation of Conditional Probabilities

N. N. Vorob'ev   ; D. K. Faddeev

Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 1, pp. 116-118

Résumé

A probability measure $\mu$ on a finite set $R$ is called interior if $\mu(a)>0$ for any $a\in R$. The set of all interior measures on $R$ is denoted by $W(R)$. Theorem. There exists a mapping $\varphi$ of $W(R)$ into Euclidean space $E$ of suitable dimension with two properties: 1. All conditional probabilities $$\mu(a|A)=\frac{\mu (a)}{\mu (A)},\quad a\in A\subset R,$$ are uniformly continuous functions $\varphi(\mu)$ on the whole set $\varphi W(R)$ in the sense of the metric on $E$. 2. The closure of $\varphi W(R)$ in $E$ is homeomorphic to the closed simplex of suitable dimension.