Lusin’s Theorem and Egorov’s Theorem

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 f be a real-valued measurable function on E. Then for each \epsilon>0, there is a continuous function g on \mathbb{R} and a closed set F\subseteq E for which \displaystyle f=g\ \text{on}\ F\ \text{and}\ m(E\setminus F)<\epsilon.

Egorov’s Theorem

Informally, “every convergent sequence of functions is nearly uniformly convergent.”

(Royden) Assume m(E)<\infty. Let \{f_n\} be a sequence of measurable functions on E that converges pointwise on E to the real-valued function f.

Then for each \epsilon>0, there is a closed set F\subseteq E for which \displaystyle f_n\to f\ \text{uniformly on}\ F\ \text{and}\ m(E\setminus F)<\epsilon.

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One Response to Lusin’s Theorem and Egorov’s Theorem

  1. Pingback: Lusin’s Theorem and Egorov’s Theorem — Singapore Maths Tuition | Mathpresso

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