Postscript III: Measurability and Projective Sets

Abstract

In this postscript, we describe two important classical problems of real analysis that could not be settled using the usual axioms of set theory: (1) The Measure Problem on extending Lebesgue measure to all of $\boldsymbol{\mathcal{P}}(\mathbf{R})$, and (2) Lusin’s Problem on properties of PCA sets and projective sets. Ulam’s analysis of Problem 1 (Measure Problem) led to large cardinals known as measurable cardinals, which, surprisingly enough, was shown by Solovay to have remarkable implications for Problem 2 (Lusin’s Problem) as well. The independence results mentioned here illustrate the prophetic nature of Lusin’s conviction that the problems of PCA and projective sets are unsolvable. This also sets up the background for Postscript IV which will describe how larger cardinals and determinacy essentially “solve” (!) Lusin’s Problem.