1. Banach algebras question 6

Here is the last my question in this theme.

I've problems with the following statements in red shapes. How can I show/explain this in the easiest way?
Where we used fact that this measure is Radon measure?

PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf

2. Originally Posted by Arczi1984
Here is the last my question in this theme.

I've problems with the following statements in red shapes. How can I show/explain this in the easiest way?
Where we used fact that this measure is Radon measure?

PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
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 $(p_n)$ converges to a limit p, and (since x is continuous) the condition $|x(p_n)-f(x)|\to0$ implies that $x(p) = f(x)$.