The classical concept of bounded completeness and its relation to sufficiency and ancillarity play a fundamental role in unbiased estimation, unbiased testing, and the validity of inference in the presence of nuisance parameters. In this short note, we provide a direct proof of a little-known result by \cite{Far62} on a characterization of bounded completeness based on an $L^1$ denseness property of the linear span of likelihood ratios. As an application, we show that an experiment with infinite-dimensional observation space is boundedly complete iff suitably chosen restricted subexperiments with finite-dimensional observation spaces are.

翻译：经典的有界完备性概念及其与充分性和辅助性的关系，在无偏估计、无偏检验以及存在讨厌参数时的推断有效性中扮演着基础性角色。本文中，我们给出了\cite{Far62}一个鲜为人知结果的一个直接证明，该结果基于似然比线性张成的$L^1$稠密性对之有界完备性进行了刻画。作为应用，我们证明：一个具有无限维观测空间的试验是有界完备的，当且仅当适当选取的、具有有限维观测空间的限制子试验是有界完备的。