How to prove a function is convex?

Question

if $V=L^p(S,\Bbb R)$ and $g \in V$, how to prove f is convex for {$f \in V : f \geq g, \mu - a.e.$}?

I understand A is convex if $x,y,\in A, (1-a)x+ay\in A$ and $0<a<1$ Do I have to prove $x,y\in S, f((1-a)x+ay)\leq(1-a)f(x)+af(y)$?