Measure Space
Table of Contents
Aka Probability Space, Measure Space, Probability Triple
A probability triple or measure space is a triple \((\Omega, \mathcal{F}, P)\), where:
- \(\Omega\) sample space is the set of all possible outcomes of the random process modelled by the probability space
- \(\mathcal{F}\) is a family of sets representing allowable events, \(\mathcal{F} \in \Omega\)
- \(P\) is a probability measure that assigns probability to events: \(P : \mathcal{F} \to [0,1]\)
With the following constraints:
- \(\Omega\) must be non empty
- The set of events \(\mathcal{F}\) must be a sigma algebra of \(\Omega\). Otherwise a probability measure can't exist.
- The probability measure must satisfy:
- \(P(\phi) = 0\), \(P(\Omega) = 1\)
For any countable sequence of mutually disjoint events $E1, E2, E3,…$
\[ P \left( \bigcup_{i} E_i \right) = \sum_i P(E_i) \]
1. Sigma Algebra
A collection \(\mathcal{F}\) of subsets of \(\Omega\), is a \(\sigma\) algebra if:
- it contains \(\Omega\)
- it is closed under complement
- it is closed under countable unions
- it is closed under countable intersections (this follows from 2 and 3)
So, an algebra closed under countable unions is a sigma algebra.
Sigma algebra is a collection of subsets of the sample space \(\Omega\) that is used to formally measure probability. A usual choice of \(\mathcal{F}\) is \(2^{\Omega}\), i.e. the set of all possible events in \(\Omega\).
We can't define probability on any arbitrary collection of subsets. For example, lookup the Vitali set.
Misc:
- the \(\sigma\) come from Greek "Summe" meaning sum, which in this context denotes union. So, \(\sigma\) algebra is called such because it is closed under union.
- Sigma algbera is also called Sigma Field. The use of "Field" is not the same as the Field in Ring Theory but rather as a type of Field of Sets.
- Proof of Property 4: \((A^c \cup B^c)^c = A \cap B\) so, closure under complement and union implies closure under intersection.