John L. Kelley

Integrals* and Products

It will be convenient to extend the domain of an integral to include certain ℝ* valued functions, and to extend the integral to an ℝ* valued functional on the larger domain. We make this extension and subsequently phrase the Beppo Levi theorem and Fatou’s lemma in this context. A more serious use of the new construct is then made in the study of product integrals and product measures.

Measures* and Mappings

A measure in the extended sense, or just a measure*, is a non-negative, countably additive, ℝ* valued function μ on a δ-ring A with μ(∅) = 0. The function on A that is 0 at ∅ and ∞ elsewhere is a measure*, each measure is a measure*, and each finite valued measure* is a measure. Classical Lebesgue measure for ℝ (see note 4.13 (i)) is the prototypical example of a measure*. A function f is integrable (or integrable*) w.r.t. a measure* μ on A iff it is integrable (integrable*) w.r.t. the measure μ 0 = μ|{A: A ∈Aand μ(A) < ∞} and in this case ∫ f dμ = ∫ f dμ 0 . Thus the integral w.r.t. classical Lebesgue measure is indentical with the integral w.r.t. Λ1.

Measurability and σ-Simplicity

We need further information on the structure of integrable functions if our theory of integration is to be conveniently usable. For example, if J on M is the Daniell extension of the pre-integral induced by a length function, must every continuous function with compact support belong to M? The answer is not self-evident, although it had certainly better be “yes”! We shall presently find criteria for integrability involving a set theoretic (measurability) requirement, and a magnitude requirement.

The Integral I μ on L 1 (μ)

This section is devoted to the construction of an integral I μ from a measure μ, to the relationships between μ and I μ (especially for Borel measures μ for ℝ), and to a brief consideration of the vector spaces L p (μ), 1 ≦ p ≦∞, associated with μ.

Pre-Measure to Pre-Integral

Each length function λ induces a rudimentary integration process as follows. If the function χ[a:b] is 1 on the interval [a:b] and 0 elsewhere, then its “integral” I λ (χ[a:b]) with respect to λ should be λ[a:b] and if f = ∑ i=1 n c i χ[a:b] then I λ (f) should be ∑ i=1 n c i λ[a i :b i ]. But is this assignment non-ambiguous? Stated in another way: does the function χ[a:b] ↦λ[a:b] have a linear extension to the vector space of linear combinations of functions of the form χ[a:b]? It turns out that this is the case, and that it is a consequence of the fact that λ has an additive extension to a ring of sets containing the closed intervals, as we presently demonstrate.

Signed Measures and Indefinite Integrals

A real (finite) valued function f on X is locally μ integrable iff μ is a measure on a δ-ring A of subsets of X and f χ A ∈ L 1 (μ)for all A in A. In this case f.μ, the indefinite integral of f with respect to μ, is the function A↦∫ A A for A in A. This function is always countably additive and hence f.μ is a measure if f is non-negative. Consequently f.μ is the difference of two measures, f + .μ and f - .μ. These two measures have little to do with each other: one of them “lives on” the set {x:f(x) ≧ 0} and the other lives on {x:f(x) < 0}.

Integral to Measure

A measure is a real valued, non-negative, countably additive function on a δ-ring A. A δ-ring is a ring A of sets such that if {A n } n is a sequence in A. then ∩ n A n ∈ A; that is, a δ-ring is a ring A that is closed under countable intersection. The family of all finite subsets of ℝ, the family of all countable subsets of ℝ, and the family of all bounded subsets of ℝ are examples of δ-rings. We observe that one of these families is closed under countable union but the other two are not.

The Category Theorems

This short chapter is concerned with the concept of category and with its application to the theory of linear topological spaces. The results include some of the most profound and most useful theorems of the subject of linear topological spaces, and are among the most important of the applications of category.

Convexity in Linear Topological Spaces

This chapter, which begins our intensive use of scalar multiplication in the theory of linear topological spaces, marks the definite separation of this theory from that of topological groups. The results obtained here do not have generalizations or even analogues in the theory of groups.