ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-24

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Counting premeasure on the finite-cofinite algebra induces counting outer measure

Example

On the finite-cofinite algebra $A_{0}$ of $N$ , let $μ_{0} (A) = ∣ A ∣$ for finite $A$ and $μ_{0} (A) = + \infty$ for cofinite $A$ . Then $μ_{0}$ is a premeasure and its induced outer measure is counting outer measure on every subset of $N$ .

Facts & Assumptions

Given: The finite-cofinite algebra and the function $μ_{0}$ in the Example.

[F1]

A premeasure on an algebra $A_{0}$ vanishes at the empty set and is countably additive whenever a disjoint sequence in $A_{0}$ has its union in $A_{0}$ . (Premeasures on algebras of sets)

[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]

Every countable algebra cover of an algebra set disjointifies inside that set into algebra members subordinate to the original cover. (A countable algebra cover disjointifies inside the covered algebra set)

[L2]

Counting measure is an outer measure on $P (X)$ and every subset of $X$ is Carathéodory measurable. (Counting measure is an outer measure for which every subset is measurable)

Verification

technique · direct

1.1F1algebra

The function is the restriction of counting measure, so a disjoint sequence whose union lies in the finite-cofinite algebra has cardinality equal to the nonnegative sum of the member cardinalities, both for a finite union and for an infinite union; hence [F1] holds.

1.2F2L1cases

If $E$ is finite, its self-cover gives induced cost at most $∣ E ∣$ , while [L1] applied to any cover of $E$ yields disjoint subordinate pieces whose total cardinality is $∣ E ∣$ , so every cover costs at least $∣ E ∣$ . If $E$ is infinite, a cover containing a cofinite member has infinite cost; a cover by finite members with finite total cardinality has finite union and cannot cover $E$ , so every cover has infinite cost.

2.1step 1.2L2∎

Step 1.2 gives the value $∣ E ∣$ for finite $E$ and $+ \infty$ for infinite $E$ , exactly the counting outer measure of [L2].

Depends on

Used by

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Dependency tree · two levels

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Sources

G. Folland, Real Analysis, 2nd ed., Section 1.4 (standard reference, not scraped)