Let $x$ and $z$ be two independent random variables. Suppose I know the following two facts:
$P_z[f(x,z) < g(x)] > 1-\delta$ uniformly for all x;
$P_x[g(x) < h(x)] > 1-\delta_h$
How can I combine these to say something about $P_x[f(x,z) < h(x)] > 1-?$ with high probability ($1-\delta$) w.r.t. $z$?