Conditional Expectations

Suppose , is a closed space of since is complete. For , the Hilbert Projection Theorem implies

where , and

The projection of onto is called the conditional expectation of with respect to .

Proposition. For all , we have , since .

The above proposition leads to a broader definition of conditional expectation.

Definition. Suppose , If is undistinguishable with over in the weak sense, i.e.

Then is called the ..., denoted by .

Theorem. Conditional Expectation Exists.
Proof. is a signed measure and absolutely continuous with respect to on .

Then the Radon-Nikodym theorem implies there is unique satisfies

Definition. Conditional Expectation with respect to r.v.
Remark.

Definition. (Conditional Expectation with respect to an event with non-zero probability.)

If , we define
Note that

Definition.