Construction of Daniell Integrals

Abstract

Daniell integrals and expectations in common use in statistics exist which are not weighted sums and are not derived from weighted sums by use of distributions or inverse distributions. This chapter considers two basic techniques for construction of Daniell integrals. The first technique, described in Section 4.1, applies to a population S and a pseudometric d on S. As in Riesz (1914) and Tjur (1980, p. 17), if S is locally compact relative to d, then any positive linear functional G on C _0d is a Daniell preintegral. Then, as in Section 2.3, H = Ix(G) is a Daniell integral. In many typical cases, the positive linear functional G on C _0d is readily constructed by using limits of weighted sums. Thus, in Section 4.1, Lebesgue (1904, 1910) integrals are developed for subpopulations of the real line. Transformations and weights are then used in Section 4.2 to construct a variety of Daniell integrals and expectations in common use. In Section 4.3, product integrals are developed and used to construct Lebesgue integrals for subpopulations of R ^T for finite populations T.