Integrable and Measurable Functions

Summary

3.1 Given a Daniell measure µ, the upper integral of a positive function with respect to Vµ is defined. A positive function is said to be µ-negligible if its upper integral is null. This allows us to define negligible sets and the notion of property true “almost everywhere”. We then prove a few important results such as Beppo Levi’s theorem (Theorem 3.1.1), Fatou’s lemma (Proposition 3.1.2), and the Riesz-Fischer theorem (Theorem 3.1.3) on the completeness of ℒ¹ (µ).

3.2 One of the main limitations of Riemann’s integral lies in the fact that it does not yield significant results when it comes to the integral of sequences or series of functions. For example, if a uniformly bounded sequence of Riemann integrable functions converges pointwise on [a, b] to a (necessarily bounded) function f, then f may not be Riemann integrable. The monotone convergence theorem (Theorem 3.2.1), Fatou’s theorem (Proposition 3.2.1), and the Lebesgue dominated convergence theorem (Theorem 3.2.2) will answer some of our questions. For a first lecture, the reader may, without loosing too much, translate “filters” into “sequences”. The next two results, continuity and differentiability of an integral with respect to a parameter (Proposition 3.2.3 and Theorem 3.2.3), are of constant use in analysis.

3.3 In this short section we focus our attention to sets rather than functions and define µ-integrable sets and µ-moderate sets.

3.4 This section is devoted to the definition of σ-measurable spaces, that is, sets endowed with a σ -ring, and the notion of measurable mapping between two such spaces. It is important to notice that this notion of measurability does not depend on a measure.

3.5 The main result of this section is the following form of Egorov’s theorem (Theorem 3.5.1): a sequence of µ-measurable mappings from Ω into a metrizable space which converges locally almost everywhere to f converges uniformly to f on the complement of an arbitrarily small integrable set. Finally, we prove that a µ-measurable function from Ω into a Banach space is integrable if and only if its upper integral with respect to Vµ is finite (Theorem 3.5.3).

3.6 The essential integral of functions is defined, as well as bounded measures.

3.7 First, we define the upper and lower integral of a positive function: f is integrable if and only if its upper and lower integrals are finite and equal. Notice that if f is integrable so is |f|: in this theory, there is no “improper integral”. We then prove Jensen’s inequality (Theorem 3.7.3) which is an important tool both in the theory of probability and in analysis.

3.8 Intuitively, an atom for a measure µ is a µ-integrable set which has no smaller proper subset (smaller and proper both in the sense of measure theory). A measure without atom is said to be diffuse; Lebesgue measure is an example of such a measure. On the other hand, a measure is said to be atomic if each nonnegligible integrable set contains an atom. The counting measure on the semiring of finite subsets of N is an example (cf. Section 6.3). If µ is atomic and f is a function from Ω into a metrizable space, f is measurable if and only if it is constant a.e. on each atom (Theorem 3.8.1).

3.9 A Daniell measure µ defines a measure µ̂ on the ring R̂ of integrable sets, called the main prolongation of µ. Similarly, µ defines a measure µ̄, called the essential prolongation of µ, on the ring R̄ of essentially integrable sets. We then study various relationships between these measures.