Definition
Closure under Complements
Let
. It is closed under complements when Whenever the family contains an event, it therefore also contains the event that it does not occur. This is one of the defining axioms of a sigma algebra.
Contrast
Every member needs its complementary partner
Let
. Compare
- Holds:
is closed under complements because its members occur in the pairs and . - Fails:
is not closed under complements because but . One missing complementary partner is enough for the axiom to fail.