Joint conditional dominated convergence
Abstract
Dominated varying observables have simultaneous almost-sure and mean absolute conditional limits along increasing information.
Let (Ω, A, μ) be an arbitrary probability space. Let F₀ ⊆ F₁ ⊆ ⋯ ⊆ A be sigma-algebras, and let M be the sigma-algebra generated by their union. Write Cⱼf = Eμ[f | Fⱼ] and C∞f = Eμ[f | M]. These are actual measurable versions of conditional expectation under the same measure μ. No relation between the input functions and the conditioning fields is required.
Theorem 1.1 (Both limits on the full rectangular tail).
Lean statement: D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/JointConditionalDominatedConvergence.tendsto_condExp_of_dominated_convergence
Proof. Machine-checked in Lean as D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/JointConditionalDominatedConvergence.tendsto_condExp_of_dominated_convergence (✓ std3). ∎
Citation. David Blackwell and Lester Dubins (1962). Merging of Opinions with Increasing Information. DOI: 10.1214/aoms/1177704456. URL: https://doi.org/10.1214/aoms/1177704456.
Commentary.
For every universe and every probability space as above, every increasing natural-indexed family of ambient sub-sigma-algebras, every sequence of ambient-measurable real functions yₙ and ambient-measurable real function y, and every integrable real function g, assume yₙ → y almost everywhere and, for every n, |yₙ| ≤ g almost everywhere. Then there is one μ-null set Z such that for every ω outside Z and every ε > 0 there are natural numbers N and J for which |Cⱼyₙ(ω) − C∞y(ω)| < ε for every n ≥ N and j ≥ J. In addition, for every ε > 0 there are natural numbers N and J such that ∫ |Cⱼyₙ − C∞y| dμ < ε for every n ≥ N and j ≥ J. Equivalently, the actual conditional expectations converge almost everywhere and in mean absolute error as the ordered pair (n,j) tends to infinity in the product order.
The null set is chosen before ε and both tail indices. The thresholds for almost-everywhere convergence may depend on ω and ε; the thresholds for the integral limit depend on ε. Both indices can increase independently, so the conclusion includes every cofinal choice of their relative speeds. It is stronger than a statement about the single diagonal or either order of iterated limits.
Intersect the countably many original domination sets with the original convergence set. On the resulting common set, |y| ≤ g and |yₙ − y| ≤ 2|g|. For each N define the actual tail error bᴺ(ω) = sup over k ≥ 0 of |y_(N+k)(ω) − y(ω)|. Each tail is measurable. On the common set, first establish its boundedness, and then obtain 0 ≤ bᴺ ≤ 2|g|, decrease in N, domination of every |yₙ − y| with n ≥ N, and bᴺ → 0. A strict ε margin before taking the supremum proves the last assertion. Suprema on the exceptional set carry no asserted order bounds.
The integrable envelope makes every bᴺ integrable, and dominated convergence gives ∫ bᴺ dμ → 0. Set qᴺ = C∞bᴺ. Conditional positivity and order make qᴺ nonnegative and decreasing almost everywhere; conditional expectation preserves its integral. The theorem that a decreasing integrable family with integrals tending to its integrable lower bound converges almost everywhere therefore yields qᴺ → 0 almost everywhere.
For every fixed N, the upward conditional-expectation theorem gives Cⱼbᴺ → qᴺ almost everywhere, and it also gives Cⱼy → C∞y. Conditional subtraction, the absolute-value inequality and order give |Cⱼyₙ − Cⱼy| ≤ Cⱼbᴺ whenever n ≥ N, almost everywhere. Intersect all these countably indexed assertions once. At a remaining point, choose N so |qᴺ| < ε/3, then choose J so the two fixed-integrand upward errors are below ε/3. The triangle inequality now controls every n ≥ N and j ≥ J on the same rectangle.
For mean absolute convergence, contraction and the integral triangle inequality give ∫ |Cⱼyₙ − C∞y| dμ ≤ ∫ bⁿ dμ + ∫ |Cⱼy − C∞y| dμ. The first term tends to zero with n by the tail-envelope argument; the second tends to zero with j by upward convergence in mean absolute error. Independent thresholds and squeezing prove the joint integral limit. This proof does not infer almost-everywhere convergence from mean convergence.
The result applies to conditional prediction for observables that improve while information grows. Even when the limit observable has information outside M, the target is its conditional mean C∞y. The same statement allows a proper constant conditioning field, arbitrary integrable envelopes with no constant bound, null-set modifications of the inputs, and arbitrary underlying probability spaces.
No adaptedness, measurability with respect to M, constant domination bound, finite-state model, Markov property, stationarity, standard Borel structure or countable generation is assumed. There is no convergence-rate conclusion. The theorem concerns increasing fields; decreasing fields and the separate normalized-density and conditional-law arguments for predictive merging are outside its claim.
This is the increasing-field part of Blackwell and Dubins, Merging of Opinions with Increasing Information, Theorem 2, page 883, equation (2). Their domination assumption is finite expectation of supₙ |yₙ|. The present assumptions imply it: on the common domination set the countable supremum is finite and bounded by |g|, so its expectation is finite. Only this implication is needed; no equivalence is assumed. The paper states both joint limits and explicitly distinguishes the weaker diagonal conclusion.