Expectations and Daniell Integrals

Abstract

The most basic measures of size used in statistical work are Daniell (1918, 1920) integrals. These integrals are generally described in the literature of real analysis and probability theory. Useful references include Royden (1988, Ch. 16), Stone (1948), Tjur (1980), and Whittle (1992). Despite the name, Daniell integrals need not have any relationship to the customary Riemann integrals of calculus. Instead, Daniell integrals are countably additive positive linear functionals on linear lattices. Countable additivity is a generalization of the finite additivity property of Theorem 1.1. For populations S and T, let a function X from S to R ^T be summable if X(s) is summable for each s in S. For a positive linear functional H on a linear lattice Ω in R ^S, let no(X*, H) be the function on T with value no(X*(t), H) for t in T. The following definition may be used.