Define conditional expectation by integral tests

Try

Suppose an observer knows only which group an outcome belongs to. Within a group, replace an integrable reading X by its probability-weighted average. The replacement must use only the available information and preserve the total reading on every observable event. These two requirements, rather than a quotient at an individual outcome, define conditional expectation.

Predict

Predict whether conditioning on an event of probability zero can be defined by dividing two probabilities. It cannot. Conditional expectation is instead an almost-surely unique measurable function characterized by integrals over an information sigma-algebra. Its values on null information atoms are not determined by those identities.

Experiment

Use a four-outcome probability model.

Outcome	Probability	X	Information group
a	1/8	0	First
b	3/8	4	First
c	1/4	2	Second
d	1/4	6	Second

The first group has probability 1/2 and weighted X total 3/2, so its conditional value is 3. The second has value 4. Verify that the replacement preserves expectation on each group and on their union. Then refine the information to distinguish a from b and observe what changes.

Understand

Let G be a sub-sigma-algebra of a probability space and X an integrable real random variable. A conditional expectation Y=E(X given G) is G-measurable, integrable, and satisfies integral_A Y dP=integral_A X dP for every A in G. This characterizes a function up to almost-sure equality, not necessarily a particular pointwise version.

For X≥0, define ν(A)=integral_A X dP on G. It is a finite positive measure and is absolutely continuous with respect to P restricted to G. Radon–Nikodym gives a nonnegative G-measurable density Y. Testing the whole space proves its integrability. For signed X, apply the construction to X⁺ and X⁻ and subtract their integrable densities.

For uniqueness, if Y and Z both work, the G-event where Y≥Z+1/k has zero integral of Y−Z, but that integral is at least its probability divided by k. Each such event is null, so Y≤Z almost surely; reverse the roles. Linearity and positivity follow from the defining identities and uniqueness. Testing the whole space gives E(Y)=E(X).

If H is contained in G, the conditional expectation of Y given H has the same integrals as X on every H-event, hence equals E(X given H). This proves the tower property. For a bounded G-measurable multiplier Z, first prove the take-out rule for indicators using intersections of G-events, then simple functions by linearity. Uniformly bounded simple approximation and dominated convergence extend it to bounded Z. Both sides are integrable because the multiplier is bounded.

On a finite information partition, the construction reduces to the weighted average on every positive-probability atom. On zero-probability atoms any value may be chosen. On continuous spaces, this abstract definition remains meaningful even when individual observed values have probability zero. A regular conditional distribution is a further object requiring its own existence hypotheses; it is not automatically supplied by a ratio formula.

Apply

For independent X and G, explain why the constant E(X) satisfies the conditional-expectation identities: independence factors the integral against each G-event, first for simple X and then by integrable approximation. Contrast this with the case where X is already G-measurable, in which the conditional expectation equals X almost surely.

Continue

Conditional expectation preserves every average visible to the chosen information, while discarding distinctions the observer cannot make. Next, convexity and Hilbert geometry show why it reduces Lp size and is the best square-integrable predictor using that information. The integral characterization remains the proof tool throughout.

Continue to View conditioning as a contraction and projection.