# A proposition from Neyman-Pearson lemma

The following proposition stems from the proof of Neyman-Pearson lemma. The result is not quite intuitive, and is there any easier proof?

Proposition 1 Let ${H, F}$ are two random variables. Consider ${V = \inf_{a\ge 0} \{x a + \mathbb{E}[ (H - a F)^+]\}}$. Suppose there exists ${\hat a}$ satisfying

$\displaystyle \mathbb{E}[F I_{\{\hat a F

then ${V = x \hat a + \mathbb{E}[(H- \hat a F)^+] = \mathbb{P}\{\hat a F.