In class we work with the next version of the Riesz theorem:
Given $X$ a compact metric space and $I$ a positive, linear and continuous functional on $C(X)$, there exists a unique Borel measure $\mu$ on $X$ for which $I(f)=\int_X f d\mu$ for every $f\in C(X)$.
I wonder if there is any standard condition on the $I$ operator, such that the measure $\mu$ is non atomic.