General nonatomic measure that cannot be expressed as an integral

Question

I read in a paper (Kingman — Poisson Processes, 2005) that:

In most cases the mean [of an inhomogenous Poisson process on a set $A$] is given in terms of the rate function $\lambda(x)$ on $S$ by

\begin{align} \mu(A) &= \int_A \lambda(x)dx\qquad\textrm{for }A > \subseteq S \end{align}

but it can be a general (nonatomic) measure on $S$.

I know this is a silly question, but what kinds of nonatomic measures cannot be expressed as in the equation?

Answer 1

The Cantor set has Lebesgue measure zero, so it is enough to find an atomless measure that puts positive measure on the Cantor set and nowhere else. Now one can identify the Cantor set with the space $\{0,1\}^\mathbb{N}$ in a measurably isomorphic way and the latter space can be endowed with the coin flipping probability measure.