Editing Conditional expectation (section)

==== Discussion ====

* This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
** The definition of <math>\operatorname{E}(X \mid \mathcal{H})</math> may resemble that of <math>\operatorname{E}(X \mid H)</math> for an event <math>H</math> but these are very different objects.  The former is a <math>\mathcal{H}</math>-measurable function <math>\Omega \to \mathbb{R}^n</math>, while the latter is an element of <math>\mathbb{R}^n</math> and <math>\operatorname{E}(X \mid H)\ P(H)= \int_H X \,\mathrm{d}P= \int_H \operatorname{E} (X\mid\mathcal{H})\,\mathrm{d}P</math> for <math>H\in\mathcal{H}</math>.
** Uniqueness can be shown to be [[almost surely|almost sure]]: that is, versions of the same conditional expectation will only differ on a [[null set|set of probability zero]].
*** Often, one would like to think of <math>\operatorname{E}(X \mid \mathcal{H})</math> as a measure on <math>\Omega</math> for fixed H. For example, it is extremely useful to claim that <math>\sum_i\operatorname{E}(X_i \mid \mathcal{H})</math> is additive for almost all H. However, this does not immediately follow because each <math>\operatorname{E}(X_i \mid \mathcal{H})</math> may have a different null set. Because countable unions of null sets are null sets, for a countable set of <math>X_i</math>, one can choose "versions" of each <math>\operatorname{E}(X_i \mid \mathcal{H})</math> with aligned null sets as to maintain additivity for almost all H. However, to align the "null sets of dysfunction" of <math>\operatorname{E}(X_i \mid \mathcal{H})</math> over all possible <math>X_i</math>, and thus treat <math>\operatorname{E}(X \mid \mathcal{H} = H)</math> as an almost surely unique measure over <math>\Omega</math> (a "regular probability measure"), we need further regularity conditions. Intuitively, to do this, we need to be able to approximate all possible <math>X_i</math> with a countable set of them. This directly corresponds to the conditions for creating a regular probability measure, which are separability and completeness.
* The ''σ''-algebra <math>\mathcal{H}</math> controls the "granularity" of the conditioning.  A conditional expectation <math>E(X\mid\mathcal{H})</math> over a finer (larger) ''σ''-algebra <math>\mathcal{H}</math> retains information about the probabilities of a larger class of events.  A conditional expectation over a coarser (smaller) ''σ''-algebra averages over more events.