Showing a result for simple functions

Question

Let $X$ be a sample from $P \in \mathcal{P}$, $\delta_{0}(X)$ be a decision rule (which may be randomized) in a problem with $\mathbb{R}^k$ as the action space, and $T$ be a sufficient statistic for $P \in \mathcal{P}$. For any Borel $A \subset \mathbb{R}^k$, define $$\delta_1(T,A)=E[\delta_0(X,A)|T]$$. Let $L(P,a)$ be a loss function. Show that $\displaystyle \int L(P,a)d \delta_1(X,a) = E[\int L(P,a)d\delta_0 (X,a)|T]$

My idea of proving this is to show that this holds for a simple function $L$ and generalize that to non-negative functions $L$ by using the conditional Montone Convergence Theorem. But I can't really show the equality for a simple function $L$. That is, if we take $\displaystyle L=\sum_{i=1}^{n} c_i \mathbb{1}_{A_i}$ can I show that the given result indeed holds true?

Answer 1

You can prove this for simple functions $L=\sum_{i=1}^nc_i\mathbb 1_{A_i}$ by using linearity of conditional expectation as follows:

$ {\displaystyle \int } \sum_{i=1}^nc_i\mathbb 1_{A_i} d\delta_1(X,a) = {\displaystyle \sum_{i=1}^n }c_i \delta_1(X,A_i)$ by definition

$\hspace{44mm}={\displaystyle \sum_{i=1}^n c_i\mathbb E\left[\delta_0(X,A_i)\;|\; T \right]}$

$\hspace{44mm}=\mathbb E\left [{\displaystyle \sum_{i=1}^n c_i\delta_0(X,A_i)\;\Bigg|\; T } \right]$ by linearity of conditional expectation

$\hspace{44mm}=\mathbb E\left [{\displaystyle \int} \sum_{i=1}^n c_i\mathbb 1_{A_i} d\delta_0(X,a)\;\Bigg|\; T \right]$

$\hspace{44mm}=\mathbb E\left [{\displaystyle \int} L\; d\delta_0(X,a)\;\Bigg|\; T \right]$ as desired.