  If $\phi$ is a submeasure satisfying an appropriate lower estimate we give a
quantitative result on the total mass of a measure $\mu$ satisfying
$0\le\mu\le\phi.$ We give a dual result for supermeasures and then use these
results to investigate convexity on non-locally convex quasi-Banach lattices.
We then show how to use these results to extend some factorization theorems due
to Pisier to the setting of quasi-Banach spaces. We conclude by showing that if
$X$ is a quasi-Banach space of cotype two then any operator $T:C(\Omega)\to X$
is 2-absolutely summing and factors through a Hilbert space and discussing
general factorization theorems for cotype two spaces.
