Proposition 18.4.8EB95710FF4D7

Proof. Set $q=P_\theta [a]_{\mathcal I}$. Projection gives $q\sim _{\mathcal I}a$, so lemma 5.38.11 gives $q\preceq a$. Since $q\leq 0$, it is a competitor for $P_\theta [a]$. Together with proposition 18.4.4, this yields

Hence the two envelopes are equal and ordinarily equivalent to $a$. Since they differ from $a$ by a bounded function, they inherit its analytic presentation. $\square$