Vol. 15, No. 3, 1965

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A characterization of conditional probability

Milton Philip Olson

Vol. 15 (1965), No. 3, 971–983

The conditional probability functions relative to a sub-σ-field 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 of an 1 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.

Mathematical Subject Classification
Primary: 60.05
Received: 5 March 1964
Published: 1 September 1965
Milton Philip Olson