if f > g on T, then so are the integrals?

this has me confused right now. anyone have any tips on how to prove in general that

if f(x) > g(x) \forall x \in T, then

\int_T f\, dx > \int_T g\, dx

it seems obvious that it's true when you plot it and look at it and there's where my trouble is. how can i get past it being "obvious" and prove it?

Counterexample: Let T = ∅.  It is obvious that f(x) > g(x) ∀ x ∈ T, f, g.  However, the integrals are zero.

However, if T ≠ ∅, try proving that ∫(f(x) - g(x))dx > 0.