Abstract
If $(X,\mathscr{A})$ is a measurable space, $(\mathscr{P}_n)_{n\in\mathbf{N}}$ is an increasing sequence of nonempty sets $\mathscr{P}_n$ of probability measures and $\mathscr{B}_n$ is a sub-$\sigma$-field of $\mathscr{A}$ which is sufficient for the statistical experiment $(X,\mathscr{A},\mathscr{P}_n)$, $n\in\mathbf{N}$, then the terminal $\sigma$-field of the sequence $(\mathscr{B}_n)_{n\in\mathbf{N}}$ contains a $\sigma$-field which is sufficient for $\bigcup_{n\in\mathbf{N}}\mathscr{P}_n$.
Citation
Jürgen HILLE. "A Heredity Property of Sufficiency." Tokyo J. Math. 20 (1) 123 - 127, June 1997. https://doi.org/10.3836/tjm/1270042404
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