TITLE: Is a measure measurable?
QUESTION [5 upvotes]: 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 $\mathbb{R}$)?

REPLY [2 votes]: 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.