lp spaces – A Question About Hölder’s Inequality

According to Wikipedia:

Theorem (Hölder’s inequality). Let ($S$, $Sigma$, $mu$) be a measure space and let $p, q ∈ left(1, inftyright)$ with $1/p + 1/q = 1$. Then for all measurable real- or complex-valued functions $f$ and $g$ on $S$, $$vertvert fgvertvert_{1} leq vertvert fvertvert_pvertvert gvertvert_{q}.$$

My question: Let $fin L^{p}(mathbb R^{m})$ and $gin L^{q}(mathbb R^{m})$. Does the preliminary that $f$ and $g$ need to be measurable still hold, or is it automatatically fulfilled?