Conditional expectation as projection

Work on a probability space and let the smaller sigma-algebra represent the available information. For an integrable real random variable X, a conditional expectation is an integrable variable measurable for that smaller sigma-algebra whose integral agrees with X on every event in it. Existence follows from the Radon–Nikodym theorem. Uniqueness is only almost sure: changing the answer on a measurable null set leaves every defining integral unchanged. Conditional expectations are therefore equivalence classes unless a particular version is chosen.

For a finite partition, the answer on a positive-probability cell is the probability-weighted average of X within that cell. It is an ordinary arithmetic average only when the conditional atom weights are equal. A cell of probability zero has no prescribed average, because the integral identity cannot distinguish any values there. Conditioning on information is consequently more general than conditioning on one event, and notation involving an event of probability zero cannot be interpreted through division by its probability.

Assume now that X is square integrable. The square-integrable variables measurable for the smaller information form a closed subspace of the Hilbert space of square-integrable variables. Conditional expectation belongs to this subspace and its residual is orthogonal to every variable in it. The defining event identity first proves this for indicator functions, then for simple functions, and finally for all square-integrable tests by approximation and Cauchy–Schwarz. The zero correlation is with every permitted test, not merely with a constant.

Expanding the squared error of any other square-integrable measurable predictor yields a Pythagorean identity: the optimal residual error plus the squared distance from that predictor to the conditional expectation. The cross term vanishes by orthogonality. Thus the optimal predictor is unique up to null sets. An integrable variable without a second moment still has a conditional expectation, but squared-error geometry may be unavailable because the proposed risks can be infinite. The L1 contraction property remains meaningful even in that case.

Conditioning an integrable variable already measurable for the available information leaves it unchanged. Averaging the conditional expectation over the whole space recovers the original expectation. If one information set is contained in another, conditioning first on the larger and then the smaller equals conditioning directly on the smaller; the same integral identities prove this tower property. Without a nesting relation, two successive conditioning operations need not simplify or commute. The sigma-algebras specify the problem as much as the variable does.

Conditional Jensen gives contraction in every finite Lp space with exponent at least one whenever the original variable lies there. For squared-integrable variables this agrees with the fact that orthogonal projection cannot increase length. As more information arrives, the best predictor changes while the tower property keeps earlier forecasts consistent with later ones in conditional mean. That observation produces the fundamental martingale example: forecasts of one fixed integrable terminal quantity as an increasing information sequence is revealed.