Sigma_ALgebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion of size.
In formal terms, a σ-algebra (also σ-field, where the σ comes from the German "Summe", meaning "sum") on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair
(
X
,
Σ
)
{\displaystyle (X,\Sigma )}
is called a measurable space.
The set X is understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die {1,2,3,4,5,6}), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to capture our intuitive ideas about how sizes combine: if there is a well-defined probability that an event occurs, there should be a well-defined probability that it does not occur (closure under complements); if several sets have a well-defined size, so should their combination (countable unions); if several events have a well-defined probability of occurring, so should the event where they all occur simultaneously (countable intersections).
The definition of σ-algebra resembles other mathematical structures such as a topology (which is required to be closed under all unions but only finite intersections, and which doesn't necessarily contain all complements of its sets) or a set algebra (which is closed only under finite unions and intersections).
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