I have been disliking the theory of surreal numbers for a while, but let's test it.
So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\infty f(x)dx\ge\int_0^\infty g(x)dx$ if $\int_0^s (f(x)-g(x))dx\ge0$ for all $s$ greater than some large $S$. And all convergent integrals are considered equal to their sums.
Using surreal numbers can you give representations of such set of divergent integrals?