1. Let (S, S,μ) be a measure-space, and let f(x) and ø(x) be two μ-integrable functions defined in S. We shall assume that f(x) is a ‘difficult’ function, of which all that is known is how to calculate f(x), generally by a rather complicated process, for any given x∈S; and that ø(x) is an ‘easy’ function, whose value for any x, and whose integral over any subset T of S belonging to the class S of μ-measurable sets, are relatively simple to obtain. We shall define the sets

and assume that

It will be of interest to consider those functions ø(x) which may be regarded as approximations to f(x) in S.