For which $g$ is this integral defined?

For which $g$ is this integral defined?

Let $f \in L^1(\Omega).$
Then $\int_{\Omega}fg$ is only defined for $g \in L^\infty(\Omega)$, am I
right? Holder's inequality tells you the integral is finite. Can I widen
the class of $g$ such that the integral is defined? Thanks.