Continuation of Conditional Probabilities

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.