Comparison of Finite Experiments

In Section 19 we dealt with general decision problems of the form $$ \bar{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{D}}} = (I,D,V) $$= (I,D,V), where I: = (Ω_I,A_I) and D: = (Ω_D,A_D) denoted measurable spaces and V a set of separately measurable functions on Ω_I × Ω_D. From now on, and for the remainder of the chapter, we shall specialize the general framework in two steps: First we shall restrict our attention to decision problems of the form <Inline>2</Inline>_k(I): = (I,D_k,V) with D_k: = {1,…,k} (k ≥1) as the decision space and the set V of all bounded, separately measurable functions on Ω_I × Ω_D as the set of loss functions. Later we shall consider decision problems of the form $$ {\bar{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{D}}}_k}\,(I): = (I,{D_k},V) $$_k (I_n) with I_n = {l, …, n} (n ≥ 1).