September 6
Russell Miller,
CUNY
A computability-theoretic proof of Lusin's Theorem
Lusin's Theorem, from real analysis, states that for every Borel-measurable function $f$ from $\mathbb R$ to $\mathbb R$, and for every $\epsilon > 0$, there exists a continuous function $g$ on $\mathbb R$ such that $\{ x\in\mathbb R~:~f(x) \neq g(x)\}$ has measure $< \epsilon$. This is proven in most introductory real analysis courses, but here we will give a proof using computability theory and computable analysis. In addition to the theorem itself, the proof will establish an effective way of producing $g$ from $f$ and $\epsilon$, and will pick out, for each $f$, the specific set of troublemakers $x$ in $\mathbb R$ that create all the discontinuities.