TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Glivenko–Cantelli theorem by bracketing entropy

Statement

Let $X_{1}, X_{2}, \dots$ be iid with law $P$ , and let $F$ be a nonempty pointwise-measurable class of measurable real functions with a nonnegative integrable envelope $F$ . Suppose $N_{[]} (ϵ, F, L^{1} (P)) < \infty$ for every $ϵ > 0$ . Then $F$ is $P$ -Glivenko–Cantelli: $\sup_{f \in F} ∣ P_{n} f - P f ∣ \to 0$ in probability. If Countable Choice is assumed, then the stronger conclusion holds that this convergence is almost sure. The terms $P_{n}$ , $P$ , and the outer-probability convention are those of Empirical, centered, and Rademacher processes, and the class notion is Glivenko–Cantelli classes.

Facts & Assumptions

Given: Iid observations with law $P$ , a pointwise-measurable class with integrable envelope, and finite $L^{1} (P)$ bracketing number at every positive scale.

[A1]

For the strong-law branch, assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) to select one finite bracketing family of width at most $1 / (2 m)$ for each integer $m \geq 1$ . The convergence-in-probability branch does not use this axiom.

[F1]

Empirical, centered, and Rademacher processes: $P_{n} f = n^{- 1} \sum_{i = 1}^{n} f (X_{i})$ and $P f = \int f d P$ .

[F2]

Covering, packing, bracketing numbers, and metric entropy: an $L^{1} (P)$ bracket $[l, u]$ has measurable endpoints, satisfies $l \leq f \leq u$ $P$ -almost surely, and has width $P (u - l)$ .

[F3]

Kolmogorov iid l1 strong law: for iid real random variables with finite first absolute moment, their sample means converge almost surely to their common mean.

[F4]

Dominated convergence: pointwise convergence dominated by one integrable envelope implies convergence of integrals.

[F5]

Basic identities for a probability measure: countable subadditivity implies that a countable union of null sets is null.

Proof

Given: The objects and hypotheses in the statement.

1.1F2F5givenconstruct

Fix $ϵ > 0$ and let $F_{0}$ be the countable pointwise-dense subclass. Choose a finite $L^{1} (P)$ bracketing cover of $F$ with widths at most $ϵ / 2$ , which is strictly below $ϵ$ ; discard brackets not meeting $F_{0}$ . For each retained bracket, choose one member of $F_{0}$ in it. The envelope and that member imply that clipping its endpoints to $[- F, F]$ preserves every inclusion in the bracket $P$ -almost surely, and cannot increase the width. Since only finitely many endpoints remain and each is bounded by the integrable $F$ , they are integrable. Countability and [F5] give one probability-one event on which all pointwise-dense-class bracket inclusions hold at every observation.

2.1F1F3F4step 1.1algebra

If $f \in [l_{j}, u_{j}]$ , then $P_{n} f - P f \leq P_{n} u_{j} - P l_{j} = (P_{n} u_{j} - P u_{j}) + P (u_{j} - l_{j})$ ; likewise $P f - P_{n} f \leq (P l_{j} - P_{n} l_{j}) + P (u_{j} - l_{j})$ . Hence $\sup_{f \in F_{0}} ∣ P_{n} f - P f ∣ \leq \max_{1 \leq j \leq m} ∣ P_{n} u_{j} - P u_{j} ∣ \lor \max_{1 \leq j \leq m} ∣ P_{n} l_{j} - P l_{j} ∣ + ϵ / 2.$ Each endpoint is integrable, so [F3] makes both finite maxima tend to zero almost surely. For arbitrary $f \in F$ , choose pointwise approximants $f_{k} \in F_{0}$ ; [F4] gives $P f_{k} \to P f$ , and the finite empirical averages give $P_{n} f_{k} \to P_{n} f$ at every sample. Thus the supremum over $F$ equals the countable supremum over $F_{0}$ samplewise. Given any target $η > 0$ , take $ϵ = η$ ; the endpoint maxima then imply $P (\sup_{f \in F} ∣ P_{n} f - P f ∣ > η) \to 0$ . No simultaneous choice of brackets across scales is used for this probability-convergence conclusion.

3.1A1F3F5step 2.1algebra∎

Now assume Countable Choice. For each integer $m \geq 1$ , choose one finite bracket family of width at most $1 / (2 m)$ . Step 2.1 with $ϵ = 1 / m$ and [F3] give a probability-one event on which ${lim sup}_{n} \sup_{f} ∣ P_{n} f - P f ∣ \leq 1 / (2 m)$ . By [F5], the countable intersection of these events still has probability one. On it the nonnegative supremum has limsup at most $1 / (2 m)$ for every $m$ , hence converges to zero. This is strong $P$ -Glivenko–Cantelli convergence as defined in Glivenko–Cantelli classes.

Depends on

Used by

Dependency tree · two levels

42 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

Jon A. Wellner, Empirical Processes: Theory and Applications, Chapter 1, §6, Theorem 6.1 (standard reference, not scraped)
David Pollard, Empirical Processes: Theory and Applications, Section 8 (standard reference, not scraped)