Let X be an uncountable set, let m be the collection of all sets E in X such that either E or E^c is at most countable, and define M(E) = 0 in the first case, and M(E) = 1 in the second case. ( m here is sigma algebra in X).
Here is the solution:
The complement E^c and E is at most countable.
So it is union of open intervals whose lengths add up to 1. (relative to ...
This shows how to describe integrals of measurable functions.