While studying double integrals, a student is certainly presented with the following property:
Property. Given two functions $f$ and $g$ that are integrable over the rectangular region $R$ (region stands for $R$ being open, connected, nonempty subset of $\mathbb R^2$), if $f(x,y) \leqslant g(x,y)$ for every $(x,y)$ in $R$, then $$ \iint_R f(x,y) dx \, dy \leqslant \iint_R g(x,y) dx \, dy. $$
With some further thoughts on this property, a student can try to relate this (somehow) with convergence of double integrals. I have searched in the web about the expression 'double integral convergence', but not much results arise.
For example, let's say we're dealing with a non-pretty looking function $g.$ Then, one might say that it would be easier to find a function $f$ that satisfies the conditions of the property above to conclude something about the double integral of $g$, over $R$.
One interesting situation would be - What happens when we find a function $f$ such that
$$ \iint_R f(x,y) dx \, dy = \pm \infty \quad ?$$
In other words, what happens in the case where the integral above is divergent ? Can we conclude that the integral of $g$ over $R$ is also divergent? Does this result directly from the property stated?