A measure space $(X,\mathfrak{M},\mu)$ is decomposable if $X$ is a disjoint union of measurable subsets, $X=\bigcup_{i\in I}X_{i}$, with $\mu(X_{i})<\infty$ for all $i$, and $\mu(A)=\sum_{i\in I}\mu(A\cap X_i)$ for every measurable set $A$ of finite measure.
It is easy to see that a $\sigma$-finite measure space is decomposable.
Question: What is an example of a decomposable measure space that is not $\sigma$-finite?
A `simplest possible' example would be most welcome.