TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Basic algebra and order properties of conditional expectation

Statement

Assume AC for existence. For real $X, Y \in L^{1} (P)$ and $a, b \in R$ , $E [a X + b Y ∣ G] = a E [X ∣ G] + b E [Y ∣ G]$ . Conditional expectation is positive, preserves order and constants, satisfies $E (E [X ∣ G]) = E X$ , and $∣ E [X ∣ G] ∣ \leq E [∣ X ∣ ∣ G]$ almost surely. Also $X < Y$ almost surely implies $E [X ∣ G] < E [Y ∣ G]$ almost surely.

Facts & Assumptions

Given: AC, a probability space, a sub-sigma-algebra G, real integrable X,Y and real scalars a,b; for the strict clause assume X<Y almost surely.

[F1]

Under AC the conditional class exists and each version has the defining event integrals. (Conditional expectation as an ae class)

[F2]

Versions with the same defining data agree almost surely. (Conditional expectation is unique almost surely)

[F3]

Integrability and integrals are preserved by finite linear combinations. (The Lebesgue integral is linear on $L^{1} (μ)$ )

[F4]

A nonnegative measurable function has zero integral exactly when it is zero almost everywhere. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere)

[F5]

Linear combinations, absolute values and discrepancy sets are measurable. (Closure properties of measurable functions used by the integral)

Proof

technique · direct

1.1

Choose versions $U = E [X ∣ G]$ and $V = E [Y ∣ G]$ . The function $a U + b V$ is $G$ -measurable and integrable. For every $A \in G$ , $\int_{A} (a U + b V) = a \int_{A} X + b \int_{A} Y = \int_{A} (a X + b Y)$ . It is a version of the left side, so uniqueness proves linearity.

F1F2F3F5

2.1

If $X \geq 0$ almost surely, on $A = {U < 0} \in G$ we have $0 \leq \int_{A} X = \int_{A} U \leq 0$ . Thus $\int_{A} (- U) = 0$ , and [F4] gives $P (A) = 0$ . For $X \leq Y$ apply this to $Y - X$ and use linearity; this proves order preservation.

step 1.1F1F4F5

3.1

The constant function $c$ is integrable, $G$ -measurable, and has its own event integrals, so $E [c ∣ G] = c$ by uniqueness. Testing $A = Ω$ in [F1] gives $E U = E X$ . Finally $- ∣ X ∣ \leq X \leq ∣ X ∣$ and steps 1.1–2.1 give $- E [∣ X ∣ ∣ G] \leq U \leq E [∣ X ∣ ∣ G]$ . Hence $∣ U ∣ \leq E [∣ X ∣ ∣ G]$ and $E ∣ U ∣ \leq E ∣ X ∣$ .

step 1.1step 2.1F1F2

4.1

If $W = Y - X > 0$ almost surely, let $T = E [W ∣ G] \geq 0$ almost surely by step 2.1. The event $A = {T = 0}$ has $\int_{A} W = \int_{A} T = 0$ , so $W 1_{A} = 0$ almost surely by [F4]. Since $W > 0$ off a null set, this forces $P (A) = 0$ . Together with $P (T < 0) = 0$ this gives $T > 0$ almost surely; linearity identifies $T = V - U$ .

step 1.1step 2.1F1F4∎

Source notes

Durrett Lemma 4.1.1 and Theorem 4.1.9(a)–(b), printed pp.206,210–211; van der Vaart Lemma 1.9(i),(iii),(iv), printed p.4. The strict almost-sure statement is derived by the zero-event argument, not attributed to a counterexample.

Depends on

Used by

Conditional cauchy schwarz inequality Corollary
Conditional lp contraction Corollary
Conditional variance decomposition Corollary
Nonnegative predictable transforms preserve submartingale gains Corollary
Second moment is the expected predictable quadratic variation Corollary
Submartingale doob decomposition has increasing compensator Corollary
An adapted process need not be a martingale Counterexample
Conditioning does not preserve strict inequalities Counterexample
Conditional expectation for nonnegative variables Definition
Predictable quadratic variation in discrete time Definition
Conditioning an independent sum on one summand Example
Law of total variance Example
Likelihood ratio martingale Example
Partial sums of independent centered variables are a martingale Example
Polya urn proportion martingale Example
Square of a martingale minus quadratic compensator Example
Conditional variance is well-defined and has the second-moment formula Lemma
Martingale differences are orthogonal in l2 Lemma
Multistep martingale characterization Lemma
Bounded predictable transforms preserve martingales Theorem
Conditional expectation is the l2 orthogonal projection Theorem
Conditional fatou and dominated convergence Theorem
Conditional jensen inequality Theorem
Conditional monotone convergence Theorem
Convex functions of martingales are submartingales Theorem
Doob decomposition of an integrable adapted process Theorem
Martingales and martingale differences correspond Theorem
Square minus predictable quadratic variation is a martingale Theorem
Taking out what is known Theorem
Uniform integrability of conditional expectations of one variable Theorem

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Durrett, Probability: Theory and Examples, 5th ed. (standard reference, not scraped)