Folland doesn't required any prior familiarity with Lebesgue integration as long as the reader is willing to accept a few results (e.g., Lebesgue dominated convergence theorem) without proof.
If you want, you probably can get Appel through inter-library loan. Although not as rigourous as Folland, it is a very interesting book,
and it is still much more rigourous than many physics books. Its treatment of Green's functions starts with a pedagogical example of the Green's functions for a driven harmonic oscillator (I think; out of town right now so I'm not sure) that is basic, but illuminating.