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Integrals of measurable functions

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).

The Questions is :
Describe the integrals of the corresponding measurable functions.

Solution Preview

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 ...

Solution Summary

This shows how to describe integrals of measurable functions.