Measurable and measure spaces, extended Borel sets

The discussion up to now has been primarily concerned with the construction and properties of measures on σ-rings. There was some advantage (with a little added complication) in preserving the generality of consideration of σ-rings, rather than σ-fields during this construction process (cf. preface). In this chapter we prepare to use the results obtained so far to develop the theory of integration of functions on abstract spaces. From this point it will usually be convenient to assume that the basic σ-ring on which the measure is defined is, in fact, a σ-field. This will avoid a number of rather fussy details, and will involve negligible loss of generality for integration.

The basic framework for integration will be a space X, a σ-field S of subsets of X, and a measure μ on S. The triple (X, S, μ) will be referred to as a measure space. When μ(X)=1, μ will be called a probability measure. Probabilities are studied in depth from Chapter 9, though also occasionally appear earlier as special cases.

In most of this chapter we shall be not concerned at all with the measure μ, but just with properties of functions and transformations defined on X, in relation to S. To emphasize this absence of μ from consideration, the pair (X, S) will be referred to as a measurable space.