Suppose there are two non-negative random variables X and Y with their cumulative distribution functions $F_X$ and $F_Y$. Further, we know $F_X(t) \leq F_Y(t)$ for all $t>0$. Then, for a non-negative function $\phi(t)$, do we always have $$\int_0^\infty\phi(t)dF_X(t) \leq \int_0^\infty\phi(t)dF_Y(t)$$
For continuous case, I also tried to show density functions $f_X(t) \leq f_Y(t)$, but do not know how. My feeling is this is not right. Since if PDFs cross once, one CDF is always greater or equal to the other one.
Finally, can we replace $F_X$ and $F_Y$ to any measures?
I saw the proof somewhere while I cannot remember which reference it is.