I am reading a proof in which I do not understand the following claim:
Let $K$ be a compact metric space and let $M(K)$ be the set of signed Borel measures on $K$. Then the set $P(K)$ of positive Borel measures on $K$ is given by $$ P(K) = \bigcap_{f \in C(K), \, \, f \geq 0} \bigg\{ \mu \in M(K) \, \bigg| \int_K f \, d \mu \geq 0 \bigg\}.$$
The $( \subseteq)$ side is trivial. For the reverse inclusion, since any indicator function of an open set can be approximated by continuous functions, it follows that $$ \bigcap_{f \in C(K), \, \, f \geq 0} \bigg\{ \mu \in M(K) \, \bigg| \int_K f \, d \mu \geq 0 \bigg\} \subseteq \big\{ \mu \in M(K) \, \, \big| \, \, \mu (U) \geq 0, \text{ for any open set } U \big\}.$$ But the problem is that for general signed measures, the collection $$ \big\{ \mu \in M(K) \, \, \big| \, \, \mu (U) \geq 0, \text{ for any open set } U \big\}$$ does not necessarily form a d-system. Hence, I am not sure how to complete the proof. Any ideas?