Are all measures Lebesgue-Stieltjes measures?

Question

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?

Answer 1

There are many other measures. For example, the counting measure: $\mu(A)$ is the number of elements of $A$, with $\mu(A)=\infty$ if $A$ is infinite. This is not a Lebesgue-Stieltjes measure. Neither are the Hausdorff measures $\mathcal H^d$ with $0<d<1$. Indeed, all of these measures have $\mu([a,b])=\infty$ whenever $a<b$, which is something a Lebesgue-Stieltjes measure cannot satisfy.

A Borel measure on $\mathbb R$ is a Lebesgue-Stieltjes measure if and only if it is regular; equivalently, if it is finite on bounded sets. See Real Analysis by Royden, section 12.3.

---

Additional remark from comments: if $\mu$ is a finitely additive measure that is finite on bounded sets, then the $\sigma$-additivity of $\mu$ is equivalent to its CDF being right-continuous. One direction is here, the other direction is here.

Answer 2

Consider the Dirac measure.

$$\delta_a(E)= 1\text{ if }a\in E, 0\text{ otherwise}$$

Or what about the measure that is zero on the empty set and infinity otherwise.

Answer 3

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...