Stefano Gentili

1st ed., 2020, 463 pp.

Print ISBN 978-3-030-54939-8

Online ISBN 978-3-030-54940-4

eBook  – Softcover Book






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The text contains detailed and complete proofs and includes instructive historical introductions to key chapters. These serve to illustrate the hurdles faced by the scholars that developed the theory, and allow the novice to approach the subject from a wider angle, thus appreciating the human side of major figures in Mathematics. The style in which topics are addressed, albeit informal, always maintains a rigorous character. The attention placed in the careful layout of the logical steps of proofs, the abundant examples and the supplementary remarks disseminated throughout all contribute to render the reading pleasant and facilitate the learning process. The exposition is particularly suitable for students of Mathematics, Physics, Engineering and Statistics, besides providing the foundation essential for the study of Probability Theory and many branches of Applied Mathematics, including the Analysis of Financial Markets and other areas of Financial Engineering.

Abstract of Chapter 1 – Round-Up of Topology
The concepts of topology and of topological space took shape in the mid 1800s with the study of R and more generally Euclidean space, together with the properties of continuous maps on such spaces. Associated to the notion of topological space is the primitive concept of open set, from which the concept of neighbourhood descends, and then closed sets, closure, interior, frontier and boundary points, limit points and isolated points, etc.. The purpose of this Chapter is to present this preliminary matter.

Abstract of Chapter 2 – Types of Sets 
From the concepts of closed set and limit points descend those of derived and perfect sets. The class of compact sets is obtained from closed and (totally)
bounded sets. Moreover, from boundary sets descend the classes of nowhere dense and dense sets. We shall discuss these topics in the context of complete metric spaces, and finally prove Baire’s category theorem and deduce its main consequences.

Abstract of Chapter 3 – Borel sets in R
Open and closed subsets of R are but special families of sets in a larger class, that of Borel sets. In this Chapter with the aid of the principle of transfinite induction, we will obtain the main properties of Borel classes and uncover the hierarchical structure. These results, together with Baire’s category theorem and
its consequences, will help us understand the link existing between F σ and G δ Borel sets and the main types of subsets of R. All of this will also lead to characterize the discontinuity set of real functions using F σ Borel sets.

Abstract of Chapter 4 – Baire Functions on R
The mathematician Peter Gustav Lejeune Dirichlet, while striving to make Fourier theory rigorous and thus prove the conjecture that any map can be repre-
sented as a Fourier series, in 1829 gave sufficient conditions for said representation to exist. In the attempt to weaken these conditions and thus extend the reach of the theorem, Dirichlet first of all examined functions with infinitely many discontinuities. In doing that he came up with the celebrated function that nowadays bears his name

At the time, the Dirichlet function not only seemed to not possess an analytical expression; it could even be represented graphically. The question of which
functions (à la Dirichlet) might be represented analytically was first posed by the Italian mathematician Ulisse Dini in 1879, and subsequently addressed by a young René Baire in his 1898 PhD thesis. The main purpose of this Chapter is to present Baire’s theory of analytically representable functions.

Abstract of Chapter 5 – Borel Functions and Baire Functions
The classification of Baire functions of section 4.2 can be related to the various types of Borel subsets of R from Chapter 3, and the Lebesgue-Hausdorff theorem does exactly that. Its proof requires a number of preliminary considerations.

Abstract of Chapter 6 – Semi-algebra and Algebra of Sets
According to Lebesgue, a good measure theory should satisfy a number of reasonable properties, one of which is that the measure of a finite or countable
number of pairwise-disjoint sets equals the sum of their measures. Yet, Lebesgue measure fulfills this reasonable property only if we restrict it to special collections of sets, called σ-algebras. To obtain such an important and complex class of sets we have to start from the more simple classes, called semi-algebras and algebras of sets. The following Chapter will be devoted to these important classes of sets.

Abstract of Chapter 7 – Monotone Classes and σ-Algebras
σ-algebras are collections of sets of pivotal importance in measure theory and integration. Compared to other families, they possess a further, decisive, property: they are closed under countable unions.

Abstract of Chapter 8 – Set Functions and Measure
The aim of measure theory is to cover an arbitrary linear set of points E with a suitable collection of intervals and extract from their lengths a measure for E.
For this purpose Peano and Jordan covered the set to be measured with a finite – albeit large – number of partial intervals, whereas Lebesgue decided to employ as covering for E a countable family of Borel sets, whose theory had been in the meantime developed by Emile Borel (1898). This choice gave rise to more elaborate notions, but in return many sets, that at first were not measurable according to the Peano-Jordan theory, now became measurable .
In this Chapter, before we undertake measure theory on completely additive families, we shall examine the theory on finitely additive families. This approach,
apart from respecting the historical evolution of the subject, allows to understand which theoretical developments facilitated the passage from the measure theory of Peano-Jordan to the more refined theory of the Lebesgue measure.

Abstract of Chapter 9 – The Lebesgue Measure
It was mentioned that to set up a theory of measure capable of measuring the greatest number of subsets in a given space, Lebesgue started with their outer
measure m ∗ , dropped the property of invariance under complements and as counterpart defined inner measures in terms of outer measures of complements.
Moreover, in order to satisfy suitable conditions for a measure (such as complete additivity), Lebesgue restricted the domain of the set function m ∗ to the class of Lebesgue measurable sets, so the outer measure m ∗ will be called the Lebesgue measure m for such sets.

Abstract of Chapter 10 – Measurable Functions
For a function f to be measurable the pre-image of any measurable set in the codomain must be measurable. We met a similar situation when discussing the
Lebesgue-Hausdorff theorem, and more specifically when we defined the class of Borel functions (definition 5.1.1). On the basis of that definition and the subsequent discussion about measurable sets, we can say Borel functions represent a special instance of family of measurable functions. Now the time has come to address the matter from a broader angle.

Abstract of Chapter 11 – The Lebesgue Integral
To determine the area of the region below the graph of a map, what one usually does is divide the base, i.e. the interval of integration, in a suitable number of
subintervals. Then for each interval we choose one of the infinitely many values that f(x) attains on it. The roundoff errors made (both up and down) in doing so tend to become smaller, until they vanish, as the subintervals shrink to 0. This is the typical logic of a definition of integral such as the one adopted by Cauchy, which was justified under the assumption that the map f to be integrated is continuous.
Riemann wanted to set up a general theory to include totally discontinuous maps as well and, although his aim was different from Cauchy’s, he adopted a
similar construction. In this way he managed to capture the integrability of only certain discontinuous maps, certainly not the majority. Indeed Henri Lebesgue, in the 1926 paper “Sur le développment de la notion d’intégrale”, wrote that when Riemann’s method would work for some discontinuous functions it was by sheer accident.
Lebesgue therefore scrapped Riemann’s method altogether, considering it inadequate in view of generalizations, and adopted instead Cauchy’s guideline that
close values of f(x) should be gathered and multiplied by the measure of the corresponding set along the x-axis. For this purpose Lebesgue, in order to group the values f(x) by proximity, divided into a finite number of subintervals not the interval of integration, but the interval of range values between the infimum and supremum of f. Thus over each subinterval the values f(x) are certainly near one another, and we can make them as near as we want by refining the partition, i.e. by increasing the number of subintervals. Given the discontinuities of f, the pre-image of each subinterval in the codomain may not be one interval. On the contrary it might be scattered over the entire domain of integration, to form a complicated linear set of points. In this way the subintervals I i appearing in Riemann’s definition, whose measure is elementary, are replaced by sets A i , which may be rather complicated. Roughly, we may say that a map f is Lebesgue integrable if every set that f projects on the horizontal axis is measurable. Within this framework the study we have made of so-called measurable functions is fully justified.

Abstract of Chapter 12 – Comparing Notions of Integral
This Chapter is devoted to showing the Lebesgue integral is an extension of the Riemann integral. Indeed, although Riemann’s definition of the integral is
simple and appealing, it turns out to have limitations which make a more general concept of the integral desirable. Lebesgue’s theory of integration sorts out all main defects of Riemann’s definition. Moreover as we will see the corresponding space of Lebesgue summable functions is the completion of the space of Riemann integrable functions. In particular, the former contains special dense subspaces of functions such as: the space of continuous functions, the space of smooth functions and more, all of which play an important role in the theory of Fourier transforms.

Abstract of Chapter 13 – Functions with Bounded Variation and Absolutely Continuous Functions
The Riemann integral can be considered an evolution of Cauchy’s integral, in that certain functions that are not integrable according to Cauchy become integrable in Riemann’s theory. At the same time, alas, in the new framework integration is no longer the inverse operation to differentiation. Thus the fundamental theorem of calculus, in the version for continuous maps proved by Cauchy, loses its status of calculus’ highest pinnacle and becomes a mere special case of a much bigger picture.
The existence of continuous maps with no derivative, of integrable functions whose integral map is not differentiable and the ensuing demise of the fundamental theorem of Cauchy’s integral calculus, persuaded many mathematicians, most notably Lebesgue, to investigate the relationship between integrals and primitives. In particular Lebesgue observed that the issues with integral calculus arise when the derivative f is not bounded. Lebesgue showed that for a function f to be summable the corresponding primitive F must have bounded variation. The idea of functions with bounded variation had in the meantime been elaborated by Jordan for others reasons. Yet, Lebesgue stopped short of saying

for every x ∈ [a,b], because in that case the difference of the two sides would be a monotone map with bounded variation and zero derivative ℓ-almost everywhere on [a,x]. He proved that among functions with bounded variation, the only ones satisfying

are the absolutely continuous functions, as defined by Giuseppe Vitali. This Chapter is therefore devoted to the study of functions with bounded variation and of absolutely continuous functions.

Abstract of Chapter 14 – Foundamental Theorems of Calculus for the Lebesgue Integral
In this Chapter we shall study the differentiability properties of absolutely continuous maps. In particular we will examine the fundamental theorems of integral
calculus for Lebesgue integrable functions and the related applications, that is: integration by parts and integration by substitution. The Chapter ends with a
primer on probability theory.