**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

whenever

whenever

**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**ifand –

**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. Carath éodory’s Theorem**. Let be a set, let be an algebra on , and let

If is a countably additive map s.th.