Understanding Proposition 2.2 in Jun Shao's Mathematical Statistics I am reading Jun Shao's "Mathematical Statistics" second edition https://link.springer.com/book/10.1007/b97553.
I need help understanding the statement and the proof of proposition 2.2 on page 117:

In the statement, $\mathbb{A}$ is the space of actions for our statistical decision and $A$ is a measurable subset of $\mathbb{A}$. The detailed setting can be found on the first paragraph of page 113 section 2.3.1. The definition of (randomized) decision rule can be found on the two paragraphs right above equation (2.22).
Let us write $(\Omega, M, \mu)$ for the probability space where $X$ is defined.
I have the following questions:

*

*How exactly is $\delta_1$ defined? What is the right way of reading (2.23)? I find myself always having trouble with statements like "random variable given $T=t$". By the definition of decision rule, $\delta_0(\cdot, A)$ is a Borel function when $A \subseteq \mathbb{A}$ is a given measurable set. Since $X$ is a random variable defined $(\Omega, M, \mu)$ , we can view $Y=\delta_0(X, A)$ also as a random variable on $(\Omega, M, \mu)$. Does $E[\delta_0(X, A) \mid T=t]$ mean the conditional expectation of random variable $Y$ given the $\sigma$-algebra generated by the set $\{ T(X)=t \}$?

*If what I claim in 1 is correct, then how do I understand the first sentence in the proof? By definition of decision rule, we have to show that $\delta_1(\cdot,A)$ is a Borel function for every given $A$ and $\delta_1(t, \cdot)$ is a probability measure on $\mathbb{A}$. I can prove the latter but I am not sure about the first. It is not clear to me why this is related to sufficiency of $T$ either.

*If what I claim in 1 is correct, then how do I show the second equality? Let $C \in M$ be a $T(X)$ measurable set. It seems that I need to show:

\begin{equation*}
\int_C \int_{\mathbb{A}} L(P,a) d\delta_1(x,a)d\mu(x) = \int_C \int_{\mathbb{A}} L(P,a) d\delta_0(x,a) d\mu(x)
\end{equation*}
It is not clear to me why this holds.


*What is the importance of suddenly assuming $\mathbb{A}\subseteq \mathbb{R}^k$? Where is it used during the proof?

 A: *

*$\delta_1(t, A)$ is defined to be any function which is measurable in $t$ for each $A$, and satisfies $\delta_1(T, A) = \mathbb E[\delta_0(X, A) \mid T]$ almost surely for each $A$. (We know such a function exists by the Doob-Dynkin lemma.)


*The point about measurability we've covered in 1. The point about sufficiency is saying that you can compute the conditional expectation in $\delta_1$ without using the unknown $P$; the conditional expectation depends only on the distribution of $X \mid T$, which is known as $T$ is sufficient.


*When the loss function $L(P, a) = 1_A(a)$ for some measurable set $A \in \mathcal B^k$, your statement is equivalent to
$$\mathbb E[1_C\delta_1(T, A)] = \mathbb E[1_C\delta_0(X, A)],$$
which follows from the definition of conditional expectation. This extends to losses which are $\mathcal B^k$-simple for fixed $P$ by linearity, and then to losses which are integrable for fixed $P$ by a standard limiting argument.


*I don't think we need $\mathbb A \subseteq \mathbb R^k$ here;  maybe the author assumed it to simplify the proof in 3?
