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.
1
http://dx.doi.org/10.1090/surv/155/01
Previous Page Next Page