I'm reading the textbook Stochastic Orders by Shaked and Shanthikumar (2007), and I need some help to understand a theorem that has no proof in the book.
Suppose random variables $X$ and $Y$ have Lebesgue densities $f$ and $g$ such that $f(t)/g(t)$ increases in $t$ over the union of supports of $X$ and $Y$. Theorem 1.C.8 in the book states that if $\psi$ is any increasing function, then the likelihood ratio of $\psi(X)$ over $\psi(Y)$ is also non-decreasing.
I know that when $\psi$ is a strictly increasing function, the likelihood ratio of $\psi(X)$ over $\psi(Y)$ is simply $f\circ\psi^- / g\circ\psi^-$ and thus is increasing. But how to show the statement is true when $\psi$ is just non-decreasing?