I am reading the paper "On the Existence of Disintegrations" by Dubins and Prikry and I am having a hard time proving their Lemma 1 (whose proof is omitted). Here is a reformulation of the statement:
Fix a set $S$ and a partition $\pi$ of $S$. Let $F$ be a set of bounded real-valued functions. Without loss of generality, we identify a set with its indicator function (which is a bounded real-valued function) and assume that $\mathcal{P}(S)\subset F$. A $\pi$-kernel is a map $\kappa(.|.): F\times\pi\rightarrow\mathbb{R}$ such that, for every $h$, $\kappa(.|h)$ is an order-preserving linear functional with $\kappa(h|h)=1$ and $\kappa(f|h)\leq\sup f1_h$. Intuitively $\kappa(f|h)$ is the conditional expectation of $f$ given $h$. For each $f$, let $\tilde{f}:S\rightarrow\mathbb{R}$ be the real-valued function given by $$\tilde{f}(w)=\kappa(f|h), \text{ if $w\in h$}$$
Now consider the following two claims:
- For all $f$, if $\kappa(f|h)>0$ for all $h$, then for all positive $\epsilon$, $P\{f<-\epsilon\}=0$;
- There is an expectation on $\{\tilde{f}: f\in F\}$(an expectation $P$ on $F$ is a linear functional defined on $F$ with $Pf\leq\sup f$) such that $$Pf=Q(\tilde{f}).$$ Dubins and Prikry claim that 1 implies 2, and the proof is straightforward. However, I am having a hard time seeing why this is the case. Any hint would be greatly appreciated.