Expectation of function under bounded cdf

Question

Suppose that $F(x)$ is a cumulative distribution function such that $F(x)\geq G(x)$ for all $x$ for some function $G(x)$. If furthermore for some bounded, continuous function $h$ $$ \int \int \int_A h(x,y,z)dG(x)dG(y)dG(z)<\infty, $$ does it then also hold that $$ \int \int \int_A h(x,y,z)dF(x)dF(y)dF(z)<\infty? $$