Given: Let be a sample from , be a decision rule (which may be randomized) in a problem with as the action space, and be a sufficient statistic for . For any Borel , define
Let be a loss function. Show that
My idea of proving this is to show that this holds for a simple function and generalize that to non-negative functions by using the conditional Montone Convergence Theorem. But I can't really show the equality for a simple function . That is, if we take can I show that the given result indeed holds true?