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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Conditioning a known variable and an independent variable

Statement

Assume AC for existence. For real $X \in L^{1} (P)$ , if $X$ is $G$ -measurable, then $E [X ∣ G] = X$ . If $P ({X \in B} \cap A) = P (X \in B) P (A)$ for every real Borel $B$ and $A \in G$ , then $E [X ∣ G] = E X$ almost surely.

Facts & Assumptions

Given: AC, real integrable X and sub-sigma-algebra G; separately either X is G-measurable or its Borel events satisfy the displayed independence identity.

[F1]

Conditional classes are characterized by measurable integrable versions with all event identities. (Conditional expectation as an ae class)

[F2]

The version class is unique. (Conditional expectation is unique almost surely)

[F3]

Nonnegative Borel functions admit increasing Borel simple approximations. (Every nonnegative measurable function is the increasing limit of simple measurable functions)

[F4]

Increasing nonnegative simple limits pass through integrals. (Monotone convergence for the integral)

[F5]

Finite linear combinations and differences of integrable functions pass through the integral. (The Lebesgue integral is linear on $L^{1} (μ)$ )

Proof

technique · direct

1.1

If $X$ is $G$ -measurable it itself meets every condition for a version: integrability is assumed and every event equality is $\int_{A} X = \int_{A} X$ . Hence uniqueness gives the first identity.

F1F2

1.2

Fix $A \in G$ under the independence hypothesis. For $h = 1_{B}$ , $E [h (X) 1_{A}] = E [h (X)] P (A)$ is exactly that hypothesis. For nonnegative Borel simple $h = \sum_{j = 1}^{m} c_{j} 1_{B_{j}}$ , multiplication by $c_{j}$ and addition give the same equality. For any nonnegative Borel $h$ , compose the increasing Borel simple approximations from [F3] with $X$ and use [F4] on both sides to obtain the equality, allowing infinite values.

givenF3F4F5

2.1

Apply step 1.2 to $h (t) = t^{+}$ and $h (t) = t^{-}$ . Their expectations are finite because $X \in L^{1}$ , so subtracting yields $\int_{A} X = E X P (A)$ . The constant $E X$ is finite, $G$ -measurable and integrable, and its integral on $A$ is $E X P (A)$ . Since $A$ was arbitrary, [F1]–[F2] identify it with $E [X ∣ G]$ .

step 1.2F1F2F5∎

Source notes

Durrett Examples 4.1.3–4.1.4, printed pp.207–208; van der Vaart Examples 1.4–1.5, printed p.2. The rectangle hypothesis is extended by simple approximation explicitly, without importing a general factorization theorem.

Depends on

Used by

Nonnegative predictable transforms preserve submartingale gains Corollary
Submartingale doob decomposition has increasing compensator Corollary
A submartingale need not have increasing sample paths Counterexample
An unbounded predictable transform may lose integrability Counterexample
Conditioning an independent sum on one summand Example
Conditioning on trivial and full sigma algebras Example
Law of total variance Example
Likelihood ratio martingale Example
Partial sums of independent centered variables are a martingale Example
Product martingale from independent mean one factors Example
Square of a martingale minus quadratic compensator Example
Conditional variance is well-defined and has the second-moment formula Lemma
Multistep martingale characterization Lemma
Bounded predictable transforms preserve martingales Theorem
Conditional expectation is the l2 orthogonal projection Theorem
Doob decomposition of an integrable adapted process Theorem
Martingales and martingale differences correspond Theorem
Square minus predictable quadratic variation is a martingale Theorem

Dependency tree · two levels

24 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)