Let $X$ be a compact Haussdorff space and $C(X)$ be the space of continuous functions $f:X\rightarrow \mathbb{R}$ which carries a natural pointwise partial order. Let $\mu$ be a regular positive bounded Borel measure on $X$, so that it induces a continuous function $\mu:C(X)\rightarrow \mathbb{R}$ by $$\mu(f) = \int_X f d\mu$$ We call $\mu$ a normal measure when it preserves the suprema of positive functions: When $\{f_\alpha\}$ is a bounded net of positive functions which has a suprema $\vee f_\alpha$, then $\mu(\vee f_\alpha)=\vee \mu(f_\alpha)$.
For instance, it is known that the commutative von Neumann algebra's are precisely the $C(X)$ where $X$ is extremally disconnected and allows enough normal measures.
An example of a measure that isn't normal is a Dirac measure on a non-isolated point. I'm looking for some additional examples. In particular, examples of non-normal bounded positive regular Borel measures on a extremally disconnected compact Haussdorff space. Some references that develop the theory of normal measures would also be appreciated.