ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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.

Integrating against counting measure recovers a series

Example

On $(N, P (N), #)$ , the nonnegative Lebesgue integral is the nonnegative series: $\int f d # = \sum_{k \in N} f (k) (f : N \to [0, + \infty]) .$

Facts & Assumptions

Given: A function $f : N \to [0, + \infty]$ .

[L1]

Counting measure is a measure on $N$ (Counting measure on an arbitrary set, Counting measure is a measure).

[L2]

The nonnegative integral is defined by simple minorants (The nonnegative Lebesgue integral).

[L3]

The nonnegative integral agrees with the simple integral on simple functions, and monotone convergence passes increasing pointwise limits through the integral. (The nonnegative integral agrees with the simple integral on simple functions, Monotone convergence for the integral)

Verification

technique · direct

1.1

For each $n$ , put [L1, L2, construct] $s_{n} : = \sum_{k < n} \min {f (k), n} χ_{{k}} .$ Then $s_{n}$ is finite-valued and simple, $s_{n} ↑ f$ , and [L3] gives $\int s_{n} d # = \sum_{k < n} \min {f (k), n} # ({k}) = \sum_{k < n} \min {f (k), n}$ because each singleton has counting measure $1$ by [L1].

2.1

Applying [L3] to $s_{n} ↑ f$ gives [step 1.1, L3] ∎ $\int f d # = \lim_{n} \sum_{k < n} \min {f (k), n} .$ The diagonal truncated sums increase to the nonnegative extended series $\sum_{k \in N} f (k)$ : each is at most that series, while every fixed finite partial sum is approached from below as $n \to \infty$ . This proves the displayed identity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.15 (standard reference, not scraped)