EDIT: I have reformulated the question, since I think I may have been thinking about this in the wrong way.
The problem is to find
$$\lim_{n->\infty} \int_A \big[f(x)^n + g(x)^n \big]^{1/n} dx$$
where $g,f$ are positive measurable functions $L^1$ on a set $A$. We can easily establish
$$\int_Af(x) <\lim_{n->\infty}\int_A \big[f(x)^n + g(x)^n \big]^{1/n} < \int_{A_1}f(x) + \int_{A_2}g(x)$$ where $A_1$ is the set $x: f(x) > g(x)$ and vice versa for $A_2$. I'm thinking that I need to quantify the "slack" between the two, but I am stuck on how exactly to do this. Any ideas?