Exercise 2, § 3

Show that every numerical function $f$ belonging to $\mathscr{L}^p(X; \mathbf{R})$ is equal almost everywhere to the difference $g_1 - g_2$ of two lower semi-continuous positive functions belonging to $\mathscr{L}^p(X; \mathbf{R})$ (note that $f(x)$ is equal almost everywhere to the sum of an absolutely convergent series $\sum_{n=1}^{\infty} f_n(x)$, where the $f_n$ are continuous with compact support).