Why is vitali caratheodory theorem important? The Vitali Caratheodory theorem, as stated in Rudin's real and complex analysis, states that for nice measures $\mu$ we can approximate any $L^{1}$ function $f$ by upper lower semicontinuous $u$ and lower semicontinuous $v$ such that $u\leq f\leq v$ and
$$\int_{X} (u-v)d\mu < \epsilon$$.
I don't see why is this theorem important? Why would we want to approximate by upper and lower semicontinuous functions?

An important application of this theorem is the fundamental theorem of calculus for the Lebesgue integral (see Theorem 7.21, p.149 Rudin's Real and Complex Analysis).
Also see J. J. Koliha. A Fundamental Theorem of Calculus for Lebesgue Integration. Amer. Math. Monthly, 113(6):551–555, 2006.
