TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Assuming countable choice, a premeasure-induced outer measure is regular with generated measurable hulls

Statement

Assume the Axiom of Countable Choice. An outer measure induced by a premeasure is regular, and every set has a measurable hull in $σ (A_{0})$ .

Facts & Assumptions

Given: Countable choice, a premeasure $μ_{0}$ on $A_{0}$ , its induced outer measure $μ^{*}$ , and a subset $E \subseteq X$ .

[F1]

A measurable hull of $E$ is a Carathéodory measurable set $H \supseteq E$ with $μ^{*} (H) = μ^{*} (E)$ ; the outer measure is regular when every subset has a measurable hull. (Measurable hulls and regular outer measures)

[F2]

The set function induced by $μ_{0}$ assigns $E \subseteq X$ the infimum of $\sum_{k} μ_{0} (A_{k})$ over all countable algebra covers $E \subseteq ⋃_{k} A_{k}$ . (The outer set function induced by a premeasure)

[L1]

Assuming countable choice, the induced outer measure restricts to a complete measure on its Carathéodory sigma-algebra and to a measure on $σ (A_{0})$ extending $μ_{0}$ . (Assuming countable choice, a premeasure extends through its induced outer measure)

Proof

technique · direct

1.1F2givenchoosecases

If $μ^{*} (E) = + \infty$ , put $H = X$ . Otherwise, for each $n \in N$ , countable choice and [F2] give an algebra cover $(A_{n k})_{k}$ of $E$ with cost below $μ^{*} (E) + 1 / (n + 1)$ ; put $U_{n} = ⋃_{k} A_{n k}$ and $H = ⋂_{n} U_{n}$ .

2.1step 1.1L1algebra

In the finite case each $U_{n}$ and their intersection $H$ lie in $σ (A_{0})$ ; in the infinite case $H = X$ lies there. Thus [L1] makes $H$ Carathéodory measurable in either case.

3.1step 1.1step 2.1F1algebra∎

One has $E \subseteq H$ . In the finite case, monotonicity and each covering bound give $μ^{*} (E) \leq μ^{*} (H) \leq μ^{*} (U_{n}) < μ^{*} (E) + 1 / (n + 1)$ for every $n$ , hence $μ^{*} (H) = μ^{*} (E)$ ; in the infinite case both values are $+ \infty$ . Therefore [F1] makes $H$ a generated measurable hull, and $μ^{*}$ is regular.

Depends on

Dependency tree · 2 levels

Sources

G. Folland, Real Analysis, 2nd ed., Exercises 18(a) and 20(b) in Section 1.4