Lusin’s Theorem and Egorov’s Theorem are the second and third of Littlewood’s famous Three Principles.
There are many variations and generalisations, the most basic of which I think are found in Royden’s book.
Informally, “every measurable function is nearly continuous.”
(Royden) Let be a real-valued measurable function on . Then for each , there is a continuous function on and a closed set for which
Informally, “every convergent sequence of functions is nearly uniformly convergent.”
(Royden) Assume . Let be a sequence of measurable functions on that converges pointwise on to the real-valued function .
Then for each , there is a closed set for which