Clock Systems in Stochastic System Theory

Measure theory review: conditional expectation

Definition.Let

(Ω, F_{Ω}, P)

be a probability space,

G

a sub-

σ

-algebra of

F_{Ω}

, and

x : Ω \to R

F_{Ω}

-measurable function. Then

y : Ω \to R

is said to be a conditional expectation with respect to

G

y

G

-measurable and for all

U \in G

\int_{U} y d P = \int_{U} x d P .

Intuition: $y$ is a best approximation of $x$ given the constraint of being $G$ -measurable.

It turns out that under some reasonable assumptions, conditional expectations exist and are unique up to $P$ -almost sure equality.

Definition.Suppose that

x : Ω \to X

is an

X

-valued random variable. Then for any

U \in F_{X}

, we may ask for the conditional expectation of

1_{U} \circ x : Ω \to R

. These assemble into

P_{x} (- ∣ G) : Ω \to Δ (X)

, which we call the conditional probability distribution for $x$ given $G$ .