A formal definition, that works for square integrable random variables, is given by the following Definition Let g c F be a a-algebra and let X E L2(Q2, F, P). Then the condi- tional expectation Y=EX9 is the projection of X onto L2(Q2, g, P) Remark. By the Hilbert space projection theorem, the conditional expectation Y= minE[(X-Z) (3) Remark. The conditional expectation is itself a random variable. Its value is uncertain because it depends on precisely which events G E g actually occur. In other words, it is a(contingent )forecasting rule whose output(the forecast)depend on the content of the information revealed. For example, suppose our information set is such that we know whether the president has been shot. Then our actions may depend on whether he is or is not shot The projection-based definition is intuitively the most appealing one, but unfortu nately it only applies to square integrable stochastic variables. One way to extend the definition to merely integrable stochastic variables is to note that L is dense CI and define E [XIg] as the limit of the sequence E[,) where Xn L2 and Xn-X(in C ) Another way is the following Proposition. Let g CF be a o-algebra and let X E C(Q, F, P). Then there is an a s.(P)unique integrable random variable Z such that g-measurable andA formal definition, that works for square integrable random variables, is given by the following. Definition Let G ⊂ F be a σ-algebra and let X ∈ L2 (Ω, F, P). Then the condi￾tional expectation Y = E [X|G] is the projection of X onto L 2 (Ω, G, P). Remark. By the Hilbert space projection theorem, the conditional expectation solves Y = min Z∈L2(Ω,G,P) E £ (X − Z) 2 ¤ . (3) Remark. The conditional expectation is itself a random variable. Its value is uncertain because it depends on precisely which events G ∈ G actually occur. In other words, it is a (contingent) forecasting rule whose output (the forecast) depends on the content of the information revealed. For example, suppose our information set is such that we know whether the president has been shot. Then our actions may depend on whether he is or is not shot. The projection-based definition is intuitively the most appealing one, but unfortu￾nately it only applies to square integrable stochastic variables. One way to extend the definition to merely integrable stochastic variables is to note that L 2 is dense in L 1 and define E [X|G] as the limit of the sequence {E [Xn|G]} where Xn ∈ L2 and Xn → X (in L 1 ). Another way is the following. Proposition. Let G ⊂ F be a σ-algebra and let X ∈ L1 (Ω, F, P). Then there is an a.s. (P) unique integrable random variable Z such that 1. Z is G-measurable and 6