Equivalence of Lebesgue's Theorem and Baire Characterization Theorem - Jonald Fenecios; Abraham Racca (IndoMS, 2022)
Let X be a complete separable metric space and Y be a separable Banach space. We provide a proof of equivalence by linking explicitly the following statements: Lebesgue's Theorem. For every ε>0 there exists a countable collection of closed sets C_nbrace of X such that X=_n=1^∞C_n and ω_f( C_n)<ε for each n. Baire Characterization Theorem. For every nonempty perfect set KX, the function f|_K has at least one point of continuity in K. In fact, C(f|_K) is dense in K. Moreover, replacing ``closed'' by ``open'' in the Lebesgue's Theorem, we obtain a characterization of continuous functions on space X.