I was wondering if there is a typo for the Riemann-Lebesgue lemma here.
Should it not be written as:
... Now let $\displaystyle g_N = \sum_1^N c_n \chi_{\left({a_i, b_i}\right)} $, a simple function.
Then the same argument shows that the theorem holds for all such simple functions which are dense in $L^1$ ... Call it $g_N \left({x}\right)$, such that $\displaystyle \int \left|{f - g_N}\right| < \epsilon$.