The study of real functions has played a fundamental role in the development of mathematics over the last three centuries. The discovery of calculus by eighteenth century mathematicians, notably Newton and Leibniz, was largely due to increased understanding of the behavior of real functions. The birth of analysis is often traced to the early nineteenth century work of Cauchy, who gave precise definitions of concepts such as continuity and limits for real functions. Convergence problems while approximating real functions by Fourier series gave rise to both the Riemann and Lebesgue integrals. Cantor developed his set theory in an effort to answer uniqueness questions about Fourier series [73, 17, 138].
During this time, different techniques have been used as the theory behind them became available. For example, after Cauchy, various limiting operations such as pointwise and uniform convergence were studied, giving rise to various approximation techniques. At the turn of this century, measure theoretic techniques were exploited, leading to stochastic convergence ideas in the 1920's. Also, at about the same time topology was developed, and its applications to analysis gave rise to functional analysis.
In recent years, a new research trend has appeared which indicates the emergence of a yet another branch of inquiry that could be called set theoretic real analysis. This area is the study of families of real functions using modern techniques of set theory. These techniques include advanced forcing methods, special axioms of set theory such as Martin's axiom (MA) and proper forcing axiom (PFA), as well as some of their weaker consequences like additivity of measure and category. (See [86], [120], [57] and [7] for examples of this work.)
Set theoretic real analysis is closely allied with descriptive set theory, but
the objects studied in the two areas are different. The objects studied in
descriptive set theory are various classes of (mostly nice) sets and their
hierarchies, such as Borel sets or analytic sets. Set theoretic real analysis
uses the tools of modern set theory to study real functions and is
interested mainly in more pathological objects. Thus, the results concerning
subsets of the real line (like the series of studies on ``small'' subsets of
[100], or deep studies of the duality between measure and
category
[111, 98, 7]) are considered only remotely related to the
subject. (However, some of these duality studies spread to real analysis too.
For example, see a monograph [35].)
Set theoretic real analysis already has a long history. Its roots can be traced back to the 1920's, where powerful new techniques based on the Axiom of Choice (AC) and the Continuum Hypothesis (CH) can be seen in many papers from such journals as Fundamenta Mathematicae and Studia Mathematica. The most interesting consequences of the Continuum Hypothesis discovered in this period have been collected in 1934 monograph of Sierpinski, Hypothèse du continu [127]. The influence of Sierpinski's results (and the monograph) on the set theoretic real analysis can be best seen in the next section.
The new emergence of the field was sparked by the discovery of powerful new techniques in set theory and can be compared to the parallel development of set theoretic topology during the late 1950's and 1960's. In fact, it is a bit surprising that the development of set theoretic analysis is so much behind that of set theoretic topology, since at the beginning of the century the applicability of set theory in analysis was at least as intense as in topology. This, however, can be probably attributed to the simple fact, that in the past half of a century there were many mathematicians that knew well both topology and set theory, and very few that knew well simultaneously analysis and set theory.