A Radon measure is just a measure for which the formula on the left side of (6) defines a continuous linear functional.

In a function algebra, the spectrum of a function is just the range of the function. So the formula (6) ensures that f(x) is in the spectrum of x.

In a compact space, every sequence has a convergent subsequence. So some subsequence of the sequence converges to a limit p, and (since x is continuous) the condition implies that .