Measure, Integration and a Primer on Probability Theory

The concepts of topology and of topological space took shape in the mid 1800s with the study of ℝ and Euclidean space. Associated to the notion of topological space is the primitive concept of open set, from which neighbourhood, closed sets, closure, interior, frontier and boundary points, limit points, and isolated points all descend. This Chapter presents this preliminary matter.

From closed sets and limit points descend derived and perfect sets. The class of compact sets is obtained from closed and totally bounded sets. Moreover, from boundary sets descend nowhere dense and dense sets. These topics are discussed in the context of complete metric spaces, culminating in Baire's category theorem.

Open and closed subsets of ℝ are special families within the larger class of Borel sets. Using transfinite induction, this Chapter uncovers the hierarchical structure of Borel classes and characterises the discontinuity sets of real functions using F_σ Borel sets.

Dirichlet's 1829 work on Fourier series led him to the celebrated function bearing his name. The question of which functions might be represented analytically was first addressed by Ulisse Dini in 1879, then by René Baire in his 1898 PhD thesis. This Chapter presents Baire's theory of analytically representable functions.

The classification of Baire functions from Chapter 4 can be related to the various types of Borel subsets of ℝ from Chapter 3 via the Lebesgue–Hausdorff theorem, whose proof requires several preliminary considerations developed here.

Lebesgue measure's complete additivity requires restricting to σ-algebras. To build up to these complex classes, one must start from simpler ones — semi-algebras and algebras of sets — which are the focus of this Chapter.

σ-algebras are collections of sets of pivotal importance in measure theory and integration. Their decisive property is closure under countable unions, which makes them the natural domain for a rigorous theory of measure.

Measure theory aims to assign a measure to arbitrary linear sets. This Chapter examines the theory first on finitely additive families, respecting the historical evolution from Peano–Jordan to the more refined Lebesgue theory, before undertaking measure theory on completely additive families.

Lebesgue started with outer measure m*, dropped invariance under complements, and defined inner measures in terms of outer measures of complements. To satisfy complete additivity, he restricted m* to the class of Lebesgue measurable sets, yielding the Lebesgue measure m.

A function f is measurable when the pre-image of every measurable set in the codomain is itself measurable. Borel functions represent a special family of measurable functions, and this Chapter addresses the matter from a broader angle.

Rather than dividing the domain as Riemann did, Lebesgue divided the range, grouping values of f by proximity. A function is Lebesgue integrable if every set it projects onto the horizontal axis is measurable — justifying in full the study of measurable functions undertaken earlier.

This Chapter compares and contrasts the various notions of integral developed throughout mathematical history, placing the Lebesgue integral in its proper context relative to Cauchy, Riemann, and other approaches.

Among functions with bounded variation, only the absolutely continuous ones — as defined by Giuseppe Vitali — satisfy certain integral representation properties. This Chapter is devoted to the study of both classes.

The culminating chapter establishes the fundamental theorems of calculus in the context of the Lebesgue integral, drawing together the theory of measure, integration, and absolutely continuous functions developed throughout the book.

Includes the index, bibliography, and supplementary reference material for the volume.