ON THE LEBESGUE MEASURABILITY OF CONTINUOUS-FUNCTIONS IN CONSTRUCTIVE ANALYSIS

The paper opens with a discussion of the distinction between the classical and the constructive notions of "computable function." There then follows a description of the three main varieties of modern constructive mathematics: Bishop's constructive mathematics, the recursive constructive mathematics of the Russian School, and Brouwer's intuitionistic mathematics. The main purpose of the paper is to prove the independence, relative to Bishop's constructive mathematics, of each of the following statements: There exists a bounded, pointwise continuous map of [0, 1] into R that is not Lebesgue measurable. If mu is a positive measure on a locally compact space, then every real-valued map defined on a full set is measurable with respect to mu.