TITLE: Are all measures Lebesgue-Stieltjes measures?
QUESTION [5 upvotes]: In our lecture we ran out of time, so our prof told us a few properties about measure: He said that a measure is $\sigma$-additive iff it has a right-side continuous function that it creates. And he was not only referring to probability measures. 
After going through my lecture notes, I thought that this would imply that there can be no other measures than ones having a right-side continuous function (I think they are called Lebesgue-Stieltjes measures) as $\sigma$-additivity is a prerequisite to be a measure. So somehow, this does not fit together. Does anybody know what he could have meant here? Or was he only referring to probability measures?
Is anything unclear about my question?

REPLY [2 votes]: The statement is certainly wrong by consistency:
Right-side continuous functions are defined/only make sense for functions whos domain is the real line (actually a right-closed subset of the real line).
Thus it gives rise at most for measures over the reals (or over a right-closed subset of the reals).
However, when restricting to measures over the reals the statement remains still wrong as pointed out above.
Besides, depending on the range this yields a real measure or more generally a vector measure...