An Extension of Lusin's Theorem

Abstract

Lusin's theorem says that for any measurable real-valued function ƒ, on [0, 1] with Lebesgue measure λ for example, and e > 0, there is a set A with λ( A ) A, ƒ is continuous. Here [0, 1] can be replaced by any normal topological space and λ by any finite measure μ which is closed regular, meaning that for each Borel measurable set B, μ( B ) = sup{μ( F ): F closed, F ⊂ B ) (RAP, Theorem 7.5.2). Recall that any finite Borel measure on a metric space is closed regular (RAP, Theorem 7.1.3). Proofs of Lusin's theorem are often based on Egorov's theorem (RAP, Theorem 7.5.1), which says that if measurable functions f n from a finite measure space to a metric space converge pointwise, then for any e > 0 there is a set of measure less than e outside of which the f n converge uniformly. Here, the aim will be to extend Lusin's theorem to functions having values in any separable metric space. The proof of Lusin's theorem in RAP, however, also relied on the Tietze-Urysohn extension theorem, which says that a continuous real-valued function on a closed subset of a normal space can be extended to be continuous on the whole space. Such an extension may not exist for some range spaces: for example, the identity from {0, 1} onto itself doesn't extend to a continuous function from [0, 1] onto {0, 1}; in fact there is no such function since [0, 1] is connected.