If $P$ is a statistically complete set of distributions, the only sufficient subfield is the trivial one

Question

In this thread i solved a claim stated without proof by Bahadur that if $P=\left\{p\right\}$ is the set of all probability measures on the measurable space $\left(\Omega,\mathcal{A}\right)$, $\mathcal{A}$ is the only possible sufficient subfield. ([BAH] remark following definition 5.2)

Bahadur adds that the proposition remains true if $P$ is substituted by a complete subset. $P_0\subseteq P$ is complete in the statistical sense iff whenever $\int fdp=0$ for all $p\in P_0$, there's some $P_0$-null set $B$ (i.e. $B\in\mathcal{A}$ and $p(B)=0$ for all $p\in P_0$), such that $f\left(\omega\right)=0$ for all $\omega\in\Omega\backslash B$. ([LEH] the first paragraph of section 3, pp. 311-312).

Any help proving this modified claim 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. [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 is essentially Theorem 3.1 in 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