I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking Fubini's Theorem.
However I have only been able to find a proof of Fubini's Theorem for $\sigma$-finite measure spaces. A locally compact group is only $\sigma$-finite when it has a countable number of components. So in general this version of the proof cannot be applied. In Halmos' Measure Theory he says that a Borel Measure (Which might assume the space is locally compact) is necessarily $\sigma$-finite. But this is included as an excercise, and I have no idea how to prove it. Is it true at all?
So I am looking either for a proof of that, which would let me use the usual Fubini Theorem in $\sigma$-finite spaces -- or a separate proof of Fubini for LC Borel measures, when there is no assumption of $\sigma$-finiteness.