bibkey: blackwelldubins1962merging
authors: David Blackwell and Lester Dubins
year: 1962
title: Merging of Opinions with Increasing Information
doi: 10.1214/aoms/1177704456
url: https://doi.org/10.1214/aoms/1177704456
claim: Theorem 2 gives joint almost-everywhere and L1 conditional dominated convergence along monotone sigma-algebras.
strata_touched:
  - D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/JointConditionalDominatedConvergence
license: reference-only
triage: anchor

Joint conditional dominated convergence

David Blackwell and Lester Dubins, Merging of Opinions with Increasing Information, The Annals of Mathematical Statistics 33(3), 882–886. The primary locator is Theorem 2, page 883, equation (2).

For real random variables yₙ converging almost everywhere to y, the theorem assumes that the countable supremum of |yₙ| has finite expectation. For a monotone increasing or monotone decreasing sequence of sigma-algebras converging to a sigma-algebra M, it concludes that the actual conditional expectations of yₙ given the jth field converge to the conditional expectation of y given M, jointly in n and j, almost everywhere and in L1. The text explicitly distinguishes this joint conclusion from the weaker diagonal.

The corresponding formal result covers the increasing case on arbitrary probability spaces. Its hypotheses give an integrable real envelope g with |yₙ| ≤ g almost everywhere for each n. Countability puts all these inequalities on one full-measure set. There the nonnegative supremum of |yₙ| is finite and at most |g|; its expectation is therefore finite. This direction connects the formal hypotheses to the paper. A reverse equivalence is not used.

The proof uses decreasing measurable tail suprema of the absolute input error, conditional order and positivity, preservation of integrals, upward conditional-expectation convergence, and a rectangular epsilon argument on one common full-measure set. The L1 conclusion follows from the uniform conditional-expectation contraction estimate and fixed-integrand upward L1 convergence. The conditional-expectation, upward-convergence and dominated- convergence ingredients come from the pinned mathematical library; the joint composition is proved in the corresponding source.

Neither adapted inputs nor measurability of the limit with respect to M is required. Decreasing fields and the further conditional-law bridges needed for the paper’s predictive-merging theorem are outside this formal result. No literature novelty is claimed. The note cites mathematics only and reproduces no external implementation or paper text.

Verified locator

DOI: 10.1214/aoms/1177704456

DOI URL: https://doi.org/10.1214/aoms/1177704456

Verified original full text: http://www.dklevine.com/archive/refs4565.pdf, Theorem 2, page 883, equation (2), including the joint almost-everywhere and L1 conclusions. The theorem locator is verified against this original-paper copy; the DOI and its URL identify the publication.