Here all functions are from $[0, 1] \to \mathbb R$.

Let $f_i$ be a sequence of continuous functions such that there exists some $M > 0$ such that $|f_i| < M$ for all $i$. Does there always exist some measurable $g$ such that

$\limsup_i \int |f_i - g| = \inf_{h \in L^1} \limsup_i \int |f_i - h|$?