In Boyd and Vandenberghe's book ``Convexity Optimization'', page 79, it is said that convexity is preserved by nonnegative weighted sums, and can be extended to integrals, say, if $f(x,y)$ is convex in $x$ for each $y \in \mathcal{A}$, and $w(y) \geq 0$ for each $y\in\mathcal{A}$, then $$ \int_{\mathcal{A}} w(y)f(x,y) dy $$ is convex in $x$.
Now my problem is, is there any restrictions on $\mathcal{A}$, such that convexity is preserved? For example, if the integral limit contains functions of $x$, that is, if the boundary of $\mathcal{A}$ can be expressed as $h(f(x,y))$, is convexity still preserved or under what restrictions on $h(.)$, will convexity be preserved?
By a comparison, Kalin (1968) terms a total positivity'' to a concavity preserving rule. I don't know how to understand this and istotal positivity'' of the density function related to my question?