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
Definition 3. A probability space is a triple where is the sample space, is the family of events (i.e. an event is an element of ) which we assume to be a -field and is a function that assigns probabilities to events.
Definition 4. A non-empty class of subsets of is a monotone class if for every monotone serquence .
Proposition 1. A -field is a monotone class while, conversely, a monotone algebra is a -field.
Corollary 1. Let be the power-set (the class of all subsets of ) and .Then for any -field of subsets of ,
Definition 5. A real valued function defined on is said to be a random variable if for every Borel set we have
A random variable is an element of . Thus
Definition 6. A function is said to be a measurable map from to if
Theorem 1. If and , then is measurable. A -field generated by a class of subsets (denoted ) is the smallest -field on that makes a measurable map. That is where is any other subset containing .
Proposition 2. If is the collection of all -fields that contain then
Definition 8. The Borel -field is the smallest -field containing all open intervals of . is generated by each of the following
- the collection of all closed subsets of ;
- the collection of all subintervals of of the form ;
- the collection of all subintervals of of the form .
Theorem 2. If and are measurable maps, then is a measurable map from to .
Theorem 3. If are random variables and is measurable, then is a random variable.
Theorem 4. If are random variables then so are
Definition 9. converges almost surely (a.s.) if where
Definition 10. A -system on a set is a family of subsets of stable under finite intersection:
Lemma 1. Let . Supposed and are measures on s.th. and on . Then
Corollary 1. If two probability measures agree on a -system then they agree on the -algebra generated by that -system. Example: .
Definition 11. Dynkin Class/ or system. Let be a set, and let be a collection of subsets of . Then is called a d-system (on ) if
- if and then
- if and , then
Recall means and .
Definition 12. : The intersection of all d-systems which contain the class of subsets of . Subsequently , is a d-system, the smallest d-system which contains . Furthermore
Lemma 2. Dynkin’s lemma. If is a -system, then
A d-system which contains a -system contains the -algebra generated by that -system.
Proposition 3. A collection of subsets of is a -algebra iff it is both a -system and a d-system.
Theorem 5. Carathéodory’s Theorem. Let be a set, let be an algebra on , and let
If is a countably additive map s.th.