A null hypothesis is just a specific thing you are trying to disprove. In this case, it's simply that the min of both distributions is identical.

I am assuming we know exactly one thing about the class the measures f and g come from: for every function in that class, \int_C^{C+d} f(x) dx >= h(d) for some monotonic function h(d).

The p-value is then computed in terms of h(d), since p >= h(d)^N.