= σ Algebra Notation = σ algebra uses and re-uses many common statistics [[Statistics/ProbabilityNotation|notations]]. <> ---- == Sets and Subsets == The maximal set, which in probability applications is the '''sample space''', is notated as ''Ω''. The sample space could be a discrete set, like ''Ω = {heads, tails}''. It could be a set of discrete numbers, like ''Ω = '''N''''' (all real numbers). It could be a continuous range, like ''Ω = [0,1]''. === Subsets === Subsets are usually named with calligraphic uppercase letters, but that's not exactly practical in typed notes. Capital letters will be used instead. A subset of ''Ω'' is expressed as ''A ⊆ Ω''. === Power sets === The power set of a set (''P(Ω)'') is the set of all subsets, including the empty set (''∅'') and the set itself (''Ω''). This becomes analagous to a probability function in descrete cases. === Intersections and Unions === The '''intersection''' of two sets is notated as ''A ⋂ B''; the '''union''' of two sets is notated as ''A ⋃ B''. The intersection of all subsets ''A,,i,,'' can be expressed as: {{attachment:intersection.svg}} The union of all subsets ''A,,i,,'' can be expressed as: {{attachment:union.svg}} === Complements === The '''complement''' of a subset ''A'' is notated as ''A^c^''. ---- == Properties == A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' ---- == Sigma Algebras == A '''σ algebra''' is usually named with calligraphic uppercase letters, but that's not exactly practical in typed notes. Capital letters will be used instead. A σ algebra is notated as ''A ⊆ P(Ω)''. In other words, ''A'' is a subset of the power set of ''Ω''. To qualify as a σ algebra, ''A'' also needs to satisfy three properties: * ''Ω'' is in ''A'' * ''A'' is closed upon complementation. For any subset, the complement of that subset is also in ''A''. * ''A'' is closed upon countable unions. ---- == Maps == '''Maps''' are usually named with blackboard bold letters, but that's not exactly practical in typed notes. Bold capital letters will be used instead. A map translates a (sub)set into a real number: '''''M''': A -> '''R'''''. === Probability Measures === '''Probability measures''' are the primary use of maps with σ algebras. A parallel to the functional expression of probability, ''p(A)'', is '''''P''': A -> [0,1]''. ---- CategoryRicottone