# 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?