I am looking at two seemingly same (but not quite) Riesz representation theorems:
(Wikipedia) Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the space of compactly supported continuous functions. Then a positive continuous linear functional $\Lambda : C_c(X) \to \mathbb R$ can be represented as an integral that is, $\Lambda$ corresponds to a (unique) measure $\mu$ (regular and Borel) such that $\Lambda (f) = \int_X f d \mu$ for all $f \in C_c(X)$.
(My lecture notes) Let $X$ be a locally compact, $\sigma$-compact metric space. Then a positive linear functional $\Lambda : C_c(X) \to \mathbb R$ can be represented by a (unique) measure $\mu$ (locally finite and positive) such that $\Lambda (f) = \int_X f d \mu$ for all $f \in C_c(X)$.
I can see that metric implies Hausdorff. But I find it impossible to remember all the conditions, namely, those about $X$ (just locally compact or $\sigma$-compact as well?) and those about the measure (surely every measure is positive by definition but requiring it to be Borel and regular seems to be stronger than locally finite).
My question(s): Which of these two versions is more general? Or are they the same? And how can I tell them apart, that is, remember which version comes with which assumptions on the space and the measure?
Thanks for your help.