INTRODUCTION

The roots of DescriptiveSetTheory go back to the work of Borel, Baire and Lebesgue

around the turn of the 20th century, when the young French analysts were trying

to come to grips with the abstract notion of a function introduced by Dirichlet and

Riemann. Afunctionwastobeanarbitrarycorrespondencebetweenobjects,withno

regardforanymethodorprocedurebywhichthiscorrespondencecouldbeestablished.

They had some doubts whether so general a concept should be accepted; in any

case, it was obvious that all the specific functions which were studied in practice were

determinedbysimpleanalyticexpressions,explicitformulas,infiniteseriesandthelike.

The problem was to delineate the functions which could be defined by such accepted

methodsand search for their characteristicproperties, presumablynice propertiesnot

shared by all functions.

Baire was firstto introducein his Thesis[1899] what we nowcall Bairefunctions (of

several real variables), the smallest set which contains all continuous functions and

is closed under the taking of (pointwise) limits. He gave an inductive definition: the

continuous functions are of class 0 and for each countable ordinal , a function is of

class if it is the limit of a sequence of functions of smaller classes and is not itself

of lower class. Baire, however, concentrated on a detailed study of the functions of

class 1 and 2 and he said little about the general notion beyond the definition.

The first systematic study of definable functions was Lebesgue’s [1905], Sur les

fonctionsrepr´ esentablesanalytiquement. Thisbeautifulandseminalpapertrulystarted

the subject of descriptive set theory.

Lebesguedefinedthecollectionof analyticallyrepresentablefunctions as thesmallest

setwhichcontainsallconstantsandprojections (x1,x2,...,xn) → xi andwhichisclosed

undersums,productsandthetakingoflimits. Itiseasytoverifythattheseareprecisely

the Baire functions. Lebesgue then showed that there exist Baire functions of every

countable class and that there exist definable functions which are not analytically

representable. He also defined the Borel measurable functions and showed that they

too coincide with the Baire functions. In fact he proved a much stronger theorem

alongtheselineswhichrelatesthe hierarchy ofBairefunctionswithanaturalhierarchy

of the Borel measurable sets at each level.

Today we recognize Lebesgue [1905] as a classic work in the theory of definability.

Itintroducedandstudiedsystematicallyseveralnaturalnotionsofdefinablefunctions

andsetsanditestablishedthefirstimportanthierarchytheoremsandstructureresults

for collections of definable objects. In it we can find the origins of many standard

tools and techniques that we use today, for example universal sets and applications of

the Cantor diagonal method to questions of definability.

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http://dx.doi.org/10.1090/surv/155/01