Definition 1. A -field is a non-empty collection of subsets of the sample space closed under the formation of complements and countable unions (or equivalently of countable intesections – note ). Hence is a -field if
Definition 2. Set functions and measures. Let be a set and be an algebra on , and let be a non-negative set function
- is additive if and, for ,
- The map is called countably additive (or -additive) if and whenever is a sequence of disjoint sets in with union in , then
- Let be a measurable space, so that is a -algebra on .
- A map is called a measure on if is countable additive. The triple is called a measure space.
The measure is called finite if
and –finite if
, () s.th.
- Measure is called a probability measure if and is then called a probability triple.
- An element of is called -null if .
- A statement about points of is said to hold almost everywhere (a.s.) if