Let $(U , \mu)$ and $(V,\nu)$ be probability spaces. Let $f$ be a convex functional on $L^1(\mu)$, i.e.
$f(tX + (1-t)Y) \leq t f(X) + (1-t)f(Y)$
for all random variables $X$ and $Y$ in $L^1(\mu)$. Is there a version of Jensen's inequality for $f$? i.e., can we say that
$E_\nu f(Z) \geq f(E_\nu (Z) )$
for sufficiently well-behaved random variables $Z$ on $U\times V$?