Let $(\Omega,\Sigma)$ be a measurable space and $\mathfrak{M}$ a family of probability measures. Suppose that exists a $\sigma$-finite measure $\mu:\Sigma \to\overline{\mathbb{R}}$ such $P\ll \mu$ for all $P\in\mathfrak{M}$. If $T$ is a sufficient statistic w.r.t. to $\mathfrak{M}$, is at least one of the following propositions true?
- There's a probability measure $\mathbb{P}_0:\Sigma\to \mathbb{R}$ such that for all $X\in\cap _{P\in\mathfrak{M}}\mathcal{L}^1(P)$ we have that $X\in\mathcal{L}^1(\mathbb{P}_0)$ and $\mathbb{E}_{\mathbb{P}_0}(X|T)=E_P(X|T)$ $P$-a.e. for all $P\in\mathfrak{M}$.
- There's a probability measure $\mathbb{P}_0:\Sigma\to \mathbb{R}$ such that for all $X\in\mathcal{L}^1(\mathbb{P}_0)$ we have $X\in\mathcal{L}^1(P)$ and $\mathbb{E}_{\mathbb{P}_0}(X|T)=E_P(X|T)$ $P$-a.e. for all $P\in\mathfrak{M}$.
Using the Theorem 2.4 which can be found at the page 38 of the book "A Graduate Course on Statistical Inference" written by Li and Babu (see this link), I was able to prove that there's a probability measure $\mathbb{P}_0:\Sigma\to \mathbb{R}$ such for all $X\in(\cap _{P\in\mathfrak{M}}\mathcal{L}^1(P))\cap\mathcal{L}^1(\mathbb{P}_0)$ we have $\mathbb{E}_{\mathbb{P}_0}(X|T)=E_P(X|T)$ $P$-a.e. for all $P\in\mathfrak{M}$
I also know that $\mathbb{P}_0 $ can be defined as $\mathbb{P}_0 :=\sum_{P\in\mathfrak{N}}c_PP$ in which $\mathfrak{N}$ is an enumerable subfamily of $\mathfrak{M}$ and $\{c_P\}_{P\in\mathfrak{N}}\subseteq (0,\infty )$ satisfies $\sum_{P\in\mathfrak{N}}c_P=1$. Besides, $\mathfrak{N}$ has the following property: if $P(E)=0$ for all $P\in\mathfrak{N}$, then $P(E)=0$ for all $P\in\mathfrak{M}$.
At least one of the proposition appears in some arguments in statistics (as you can see in this link, for instance) and in the page 41 of the book I mentioned (see the proof of the Theorem 2.5).
