Editing Conditional expectation (section)

==== Existence ====

The existence of a minimizer for <math> \min_g \operatorname{E}\left((X - g(Y))^2\right)</math> is non-trivial. It can be shown that
:<math> M := \{ g(Y) : g \text{ is measurable and }\operatorname{E}(g(Y)^2) < \infty \} = L^2(\Omega, \sigma(Y)) </math>
is a closed subspace of the Hilbert space <math>L^2(\Omega)</math>.<ref>{{cite book |last1=Brockwell |first1=Peter J. |title=Time series : theory and methods |date=1991 |publisher=Springer-Verlag |location=New York |isbn=978-1-4419-0320-4 |edition=2nd}}</ref>
By the [[Hilbert projection theorem]], the necessary and sufficient condition for
<math>e_X</math> to be a minimizer is that for all <math>f(Y)</math> in {{mvar|M}} we have
:<math> \langle X - e_X(Y), f(Y) \rangle = 0. </math>
In words, this equation says that the [[residual (statistics)|residual]] <math>X - e_X(Y)</math> is orthogonal to the space {{mvar|M}} of all functions of {{mvar|Y}}.
This orthogonality condition, applied to the [[indicator function]]s <math>f(Y) = 1_{Y \in H}</math>,
is used below to extend conditional expectation to the case that {{mvar|X}} and {{mvar|Y}} are not necessarily in <math>L^2</math>.