Let $(X, M, \mu)$ be a measure space, $q \in (0, +\infty]$ and $f,g : X \rightarrow \mathbb{C}$ in which $f \in L^{\infty} (\mu)$ and $g \in L^q (\mu)$. I want to show that $fg \in L^q (\mu)$.
For this, I showed that if $g$ is an $L^1$ function on $X$ and $f$ is an $L^ ∞$ function on $X$, then $fg$ is Lebesgue integrable. But I'm not sure if this helps me to prove the question above. So, any help and idea is definitely appreciated.
$fg$ is measurable and $\int |fg|^{q} d\mu \leq \|f\|_{\infty}^{q} \int |g|^{q}d\mu <\infty$ because $|f(x)| \leq \|f\|_{\infty}$ almost everywhere. Hence $fg \in L^{q}(\mu)$.