Effectivizing Lusin's Theorem

Abstract

Lusin's Theorem states that, for every Borel-measurable function $\bf{f}$ on $\mathbb R$ and every $\epsilon>0$, there exists a continuous function $\bf{g}$ on $\mathbb R$ which is equal to $\bf{f}$ except on a set of measure $<\epsilon$. We give a proof of this result using computability theory, relating it to the near-uniformity of the Turing jump operator, and use this proof to derive several uniform computable versions. Easier results, which we prove by the same methods, include versions of Lusin's Theorem with Baire category in place of Lebesgue measure and also with Cantor space $2^{\mathbb N}$ in place of $\mathbb R$. The distinct processes showing generalized lowness for generic sets and for a set of full measure are seen to explain the differences between versions of Lusin's Theorem.