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Measure Theory
Measure
theory is a branch of mathematics that formalizes the idea of a measure. In
measure theory, a “measure” is a mathematical function that maps each set in a collection
of “measurable sets” to a non-negative, real number. Area is an example of a
measure. Probability is another example.
Under the
precept of measure theory called “additivity,” the measure of the union of
disjoint sets is the sum of the measures of the individual sets. Under a
different precept, the measure of an empty set is nil.
The
composition of the collection of measurable sets is governed by rules. Under
these rules, if the collection contains the sets A, B, C,…, then it contains the union of these
sets. Under the same rules, if the collection contains a pair of sets A and B, then it contains the pair of set differences A – B
and B – A plus the intersection of the two sets.