Let's have integral
$$ \tag 1 I = \int\limits_{x_0}^x dy \, g(y)A(y) $$ Here $g(y)$ is positive monotonically decreased function, while $A$ is a positive definite function with $\int\limits_{x_0}^\infty A \, dx < \infty$.
May I estimate $(1)$ as $$ I \leqslant g(x_0)\int\limits_{x_0}^x A(y) \, dy $$
Provided all the integrals are well defined, I see no problem arguing that if $g(y) \leqslant g(x_0) \ \forall y \in [x_0,x)$ then $$ \int_{x_0}^x A(y) g(y) dy \leqslant \int_{x_0}^x A(y) g(x_0) dy = g(x_0) \int_{x_0}^x A(y) dy $$