Equivalence of Lebesgue's Theorem and Baire Characterization Theorem

Jonald Fenecios (Philippine Science High School Southern Mindanao Campus)
Abraham Racca (Adventist University of the Philippines)

Journal of the Indonesian Mathematical Society
VOLUME 28 NUMBER 2 (JULY 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:\\ \noindent \textbf{\textit{Lebesgue's Theorem.}} For every $\epsilon>0$ there exists a countable collection of closed sets $\left\lbrace C_n\right\rbrace $ of $X$ such that $$X=\bigcup_{n=1}^{\infty}C_n\;\;\text{and}\;\; \omega_f\left( C_n\right)<\epsilon\;\; \text{for each} \;\; n.$$ \textbf{\textit{Baire Characterization Theorem.}} For every nonempty perfect set $K\subset X$, the function $f|_K$ has at least one point of continuity in $K$. In fact, $C(f|_K)$ is dense in $K$.\\ \indent Moreover, replacing ``closed'' by ``open'' in the Lebesgue's Theorem, we obtain a characterization of continuous functions on space $X$.