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.

## Lusin’s Theorem:

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

## Egorov’s Theorem

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

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