the partial-averaging condition

Here is the second, more subtle of the two requirements that define E[X given G]. We cannot ask the conditional expectation to equal X — that would let it use forbidden information. Instead we ask for the next best thing: that it equal X on average over every block of information G is allowed to resolve. Picture pixelating a photograph at the resolution G permits: within each pixel-block you replace all the true values by their average, so detail is lost but the total brightness over every block is preserved exactly.

Formally: for every event A in G, the integral of E[X given G] over A equals the integral of X over A, i.e. E[ E[X given G] times 1_A ] = E[ X times 1_A ]. Take A to be the whole space and you get E[ E[X given G] ] = E[X], the average is unchanged. There is a second, equivalent face of the same condition that explains the name 'orthogonality': the error X minus E[X given G] is uncorrelated with every G-measurable function — E[ (X - E[X given G]) times Z ] = 0 for all bounded G-measurable Z. The leftover, the part of X that G cannot explain, is geometrically perpendicular to everything G can express.

This condition is the workhorse of proofs. The tower property, taking out what is known, and the variance-decomposition identity are all just the partial-averaging condition applied with a cleverly chosen test set A or test function Z. The misconception to avoid: partial averaging does NOT say E[X given G] equals X on any individual outcome — equality holds only after integrating over blocks. On a fine enough G the blocks shrink and E[X given G] does approach X, but on a coarse G it can be very far from X at single points while still averaging correctly.