Σ-algebra
variable {α : Type}
class Σ_algebra (X : set α) (Σ : a collection of subsets of X) :
(Σ_closed_under_complement : the complement of an element in Σ is in Σ)
(Σ_closed_under_countable_unions : ...) -- what is countable
(Σ_closed_under_countable_intersections : ...)
“The pair (X, Σ) is called a Measurable space.”
Main application: define Measure: the collection of subsets for which a given measure is defined is instance of Σ-algebra.”
“Σ-algebra in probability theory: collection of events that can be assigned probabilities.”
“When conditional expectation is involved, one is concerned with sets that represent only part of all the possible information that can be observed.
This partial information can be characterized with a smaller Σ-algebra which is a subset of the principal Σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information.”