  Markov kernels play a decisive role in probability and mathematical
statistics theories, and are an extension of the concepts of sigma-field and
statistic. Concepts such as independence, sufficiency, completeness,
ancillarity or conditional distribution have been extended previously to Markov
kernels. In this paper, the concept of conditional expectation of a Markov
kernel given another is introduced, setting its first properties. An
application to clinical diagnosis is provided, obtaining {an} optimality
property of the predictive values of a diagnosis test. In a statistical
framework, this new probabilistic tool is used to extend to Markov kernels the
theorems of {Rao-Blackwell} and Lehmann-Scheff\'e. A result about the
completeness of a sufficient statistic is obtained in passing by properly
enlarging the family of probabilities. As a final statistical scholium, a
generalization of a result about the completeness of the family of
nonrandomized estimators is given.
