The conditional probability
functions relative to a sub-σ-field ℬ are shown to constitute a vector-valued measure
on the σ-field of the probability space, and it is proved that the conditional
expectation relative to ℬ of an ℒ1 random variable X is the integral of X
with respect to this vector-valued measure. A complete characterization is
given of those vector-valued measures which are conditional probabilities.
This machinery is illustratively applied to give alternative derivations of
results of Moy and Rota on the characterization of conditional expectation
operators.
