Clock Systems in Stochastic System Theory

Measure theory review: what is a $σ$ -algebra?

Definition.A

σ

-algebra on a set

Ω

is a subset

F \subset 2^{Ω}

such that

Intution: a $σ$ -algebra represents a collection of questions that we are allowed to ask of an element of $Ω$

Example.If

Ω = R^{2}

, then there is a

σ

-algebra given by

{π_{1}^{- 1} (U) ∣ U in the Borel σ -algebra on R}

This represents the questions we can ask about a point in

R^{2}

given we know its first component