Exchangeable Sequences, Laws of Large Numbers, and the ...

Conditional Expectation
( )
{ ω : ω }
( Ω )
Let X be r.v. on probability space , S, P . Given another
σ -algebra Y ⊂ S, a r.v. Y is called conditional expectation
of X given Y Y if Y is Y
Y -measurable, that is,
∈Ω Y ≤ a ∈Y ∀a ∈ℜ
and YdP = XdP ∀A∈ .
∫ ∫ Y
A A
( Y )
( Y )
In this case, denote Y = E X | .
Roughly speaking, E X | is averaging of X to the granularity
of Y Y (if YY
Y is finite, averaging on the atoms
of Y
).