Let $(X,\Sigma,\mu)$ be a measure space and $L_p(X,\Sigma,\mu)$ be the usual space of all complex/real valued measurable functions $f$ such that $$\int_X\vert f\vert^p\;d\mu<\infty$$
My question is: when is this space Banach? First of all, does $(X,\Sigma,\mu)$ have to be $\sigma$-finite or complete? If it is not, is the $L_p$ still Banach (for any $1\le p\le\infty$)?
Thanks