I'm dealing with the Riesz representation theorem to prove that the dual $C(\mathbb{T})^*$ is (isometric to) the space of complex Borel measures on $\mathbb{T}$. On the other hand I've read that the Riesz representation theorem establishes a correspondence with regular complex Borel measures on $\mathbb{T}$.
My question is: in my case with $\mathbb{T}$ the correspondence is with complex Borel measures or with regular complex Borel measures? Or it turns out that they are the same in the space $\mathbb{T}$?.
Any help would be appreciated.
From the comments: "All Borel measures on a compact metric space are regular."