Is a measure measurable?

This could totally be a stupid question but I'm unsure: is a measure (ie positive, countable additive on a $\sigma$ algebra, 0 for the empty set) actually a measurable function (wrt to the Borel-sigma algebra on $\mathbb{R}$)?

For a measure space $(X,\mathcal{F},\mu)$ you have that

$$ \mu^{-1}((-\infty ,c])=\{A\in \mathcal{F}:\mu(A)\leqslant c\} $$

so you will need a $\sigma $-algebra defined in $\mathcal{F}$ to define the measurability of $\mu$. Well, you can define this $\sigma $-algebra using $\mu$, this will give an induced $\sigma $-algebra in $\mathcal{F}$, and we can note it by $\sigma (\mu)$.

Given a function, a sigma algebra on the domain and a sigma algebra on the codomain, you can ask whether the function is measurable with respect to the given sigma algebras. You have only specified a sigma algebra on the codomain, so we can not answer your question.

What probably confused you is the fact that the domain of a measure is a sigma algebra on some set, so we need to consider a sigma algebra on another sigma algebra - the domain of the measure - to determine whether it is measurable.