Measure theory generalizes the notion of length, area and volume to a very general topological setting. It also forms the basis of probability theory.
A -algebra is a set of sets which
- is closed under difference, if ,
- is closed under countable unions, if , and
- has a unit such that for all .
A -additive measure is a function over a -algebra taking non-negative values such that
for pairwise disjoint sets .
The closure of a -algebra under countable intersections follows from de Morgan’s law,
The empty set is the unique zero element such that for all . Further, and for all .
A -algebra forms a ring, with as addition, as multiplication, as the multiplicative identity, as the additive identity, and as additive difference.
Constructing Lebesgue Measures
It’s not entirely trivial to show that interesting measures exist, even working over the real numbers. The easiest useful measure to construct is the Lebesgue measure over the unit interval .
Start by defining the measure of a subinterval to be its length. Extend the definition to countable unions of subintervals by summation, following the definition of measure.
It remains to tidy up the some boundary cases producing zero-measure results, which isn’t entirely trivial (analysis is built on counterexamples and edge cases), but the basic idea pretty much works as is.