Set of all Events

probability-theory

Definition

Set of All Events

Let be a sample space. Its set of all admissible events is a family

that forms a sigma-algebra on . Each is an event to which a probability may be assigned. Together with a probability measure , it forms the probability space .

Why It May Be Smaller Than the Power Set

The phrase “set of all events” means all events admitted by the probabilistic model, not necessarily every subset of . In finite models one often chooses , whereas models on uncountable sample spaces commonly use a proper subset

The required closure properties are those of a [[Knowledge/Sigma Algebra|-algebra]].

Example

Recording only parity

For a six-sided die, let

The event family

records whether the outcome is even or odd but cannot express an event such as . Thus .