A Convergence Property for Conditional Expectation

On Convergence of Conditional Expectation Operators

Given an operator T : UX(Σ) → Y or T : C(H,X) → Y , one may consider the net of conditional expectation operators (Tπ) directed by refinement of the partitions π. It has been shown previously that (Tπ) does not always converge to T . This paper gives several conditions under which this convergence does occur, including complete characterizations when X = R or when X∗ has the Radon-Nikodým prope...

CONDITIONAL EXPECTATION IN THE KOPKA'S D-POSETS

The notion of a $D$-poset was introduced in a connection withquantum mechanical models. In this paper, we introduce theconditional expectation of  random variables on theK^{o}pka's $D$-Poset and prove the basic properties ofconditional expectation on this  structure.

Conditional Expectation

Let μ and λ be two positive bounded measures on the same meaurable space (Ω,F). We call μ and λ equivalent, and write μ ≡ λ, if they have the same null sets— so, if they were probability measures, the notion of “a.s.” would be the same for both. More generally, we call λ absolutely continuous (AC) w.r.t. μ, and write λ μ, if μ(A) = 0 implies λ(A) = 0, i.e., if every μ-null set is also λ-null. W...

Conditional expectation

Lecture 10 Conditional Expectation

We say that P[A|B] the conditional probability of A, given B. It is important to note that the condition P[B] > 0 is crucial. When X and Y are random variables defined on the same probability space, we often want to give a meaning to the expression P[X ∈ A|Y = y], even though it is usually the case that P[Y = y] = 0. When the random vector (X, Y) admits a joint density fX,Y(x, y), and fY(y) > 0...