Minimally Sufficient Statistic

Question

In a nutshell, an example of a necessary and sufficient statistic is given in an article, but it is not explained there why the statistic is in fact necessary and sufficient. I'd appreciate help in understanding this.

Bahadur ascribes to Lehmann & Scheffe the claim that there exists a minimally sufficient statistic on the statistical space $\left(\Omega,\mathcal{A},P=\left\{p\right\}\right)$ provided $P$ is separable under the metric $d\left(p,q\right):=\sup_{B\in\mathcal{A}}\left|p(B)-q(B)\right|$. ([BAH] last paragraph on p. 440, spilling over to p. 441.) He gives an example of a minimally sufficient statistic under these conditions, as follows.

Let $Q=\left\{q_1,q_2,\dots\right\}$ be a countable, everywhere dense subset of $P$. Choose a countable subset of $P$, $P_0=\left\{p_1,p_2,\dots\right\}$, such that every $\mathcal{A}$-$P_0$ null set (i.e. every set $B\in\mathcal{A}$ such that $p(B)=0$ for all $p\in P_0$) is also $\mathcal{A}$-$P$ null and vice-versa. The existence of $P_0$ is guaranteed by [HAL] lemma 7; alternatively [BAH] remark preceding theorem 6.1. Choose any sequence of positive, real constants that sum to $1$ and define $$\lambda_0:\mathcal{A}\rightarrow\overline{\mathbb{R}},\space\space\lambda_0(B):=\sum_i c_i p_i(B)$$ $\lambda_0$ is a probability measure on $\left(\Omega,\mathcal{A}\right)$. Set $\varphi_i:=\frac{dq_i}{d\lambda_0}$.

According to Bahadur, $T^*:=\left(\varphi_1,\varphi_2,\dots\right)$ is a minimally sufficient statistic for $P$. Unfortunately, he doesn't justify this claim, except by suggesting that "the construction is based on Theorem 6.1 and Corollary 6.1". I reproduce below these two results.

Theorem 6.1 As defined, $\lambda_0$ is a probability measure on $\mathcal{A}$ such that

1. $P\equiv\lambda_0$, i.e. each $P$-null set is a $\lambda_0$-null set and vice versa.
2. A necessary and sufficient condition that a subfield $\mathcal{A}_0$ be sufficient for the measures $P$ on $\mathcal{A}$ is that corresponding to each $p\in P$, there exists a nonnegative $\mathcal{A}_0$-measurable function $g_p$ such that $g_p=\frac{dp}{d\lambda_0}$ on $\mathcal{A}$.

Corollary 6.1 Let there be given a $\sigma$-finite measure $\lambda$ on $\mathcal{A}$ such that $P\ll\lambda$. A necesary and sufficient condition that a statistic $T$ be sufficient for $P$ is that there exist a nonnegative function $h$ on $\Omega$ and a set $\left\{g_p:p\in P\right\}$ of nonnegative functions whose domain is the image of $T$ such that

1. $h$ is an $\mathcal{A}$-measurable function.
2. For each $p$, $g_p\circ T$ is an $\mathcal{A}$-measurable function.
3. For each $p$, $\frac{dp}{d\lambda}=h\cdot g_p\circ T$ on $\mathcal{A}$.

I understand why $T^*$ is a necessary statistic (this follows readily from [BAH] theorem 6.2 and corollary 6.2 (i), which i have not reproduced here), but why is it sufficient? Any help will be appreciated.

References

1. [BAH] Bahadur, R. R. Sufficiency and Statistical Decision Functions. Ann. Math. Statist. Volume 25, Number 3 (1954), 423-462
2. [HAL] Halmos, Paul R., Savage, L. J. "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics". Annals of Mathematical Statistics, 20, 225-241 (1949)
3. [LEH] Lehmann E. L., Scheffé Henry. Completeness, Similar Regions, and Unbiased Estimation: Part I. ankhyā: The Indian Journal of Statistics (1933-1960), Vol. 10, No. 4 (Nov., 1950), pp. 305-340

Answer 1

This can be proved essentialy like theorem 6.3 in Lehmann E. L., Scheffé Henry. Completeness, Similar Regions, and Unbiased Estimation: Part I. Sankhyā: The Indian Journal of Statistics (1933-1960), Vol. 10, No. 4 (Nov., 1950), pp. 305-340