19.6.2.2 Measurable functions and sets

Problem: Show that for any measurable function \( f \) on the segment \( [0,1] \) with Lebesgue measure and any \( \varepsilon>0 \) there exists such \( C \) with \( \mu(C)>1-\varepsilon \) and a continuous function \( g \) on the segment, that \( f(x)= \) \( g(x) \) when \( x \in C \).