TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

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The interval data on $(a, b]$ determines the Borel measure uniquely

Statement

Let $μ$ and $ν$ be Borel measures on $R$ finite on compact sets in the sense of A Borel measure on $R$ that is finite on compact sets. If

$μ ((a, b]) = ν ((a, b]) for every a < b,$

then $μ (E) = ν (E)$ for every Borel set $E \subseteq R$ .

Facts & Assumptions

Given: Two Borel measures $μ, ν$ on $R$ , each finite on compact sets, and agreement of $μ$ and $ν$ on every half-open interval $(a, b]$ .

[L1]

The family of half-open intervals $(a, b]$ with $a < b$ generates the Borel sigma-algebra on $R$ . (Seven generating families for the Borel sigma-algebra on the real line)

[L2]

Measures that agree on a generating pi-system and on an increasing finite-measure exhaustion from that pi-system agree on the whole sigma-algebra. (Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system)

Proof

technique · direct

1.1L1givenalgebra

Let $P = {\emptyset} \cup {(a, b] : a < b}$ . The intersection of two nonempty members is either empty or $(\max {a, c}, \min {b, d}]$ with its left endpoint strictly below its right endpoint. Intersections with $\emptyset$ are empty, so $P$ is a pi-system. By [L1], adjoining $\emptyset$ does not change the generated sigma-algebra, and $σ (P) = B (R)$ . Both measures agree on $P$ , including $\emptyset$ .

1.2givenalgebra

For each $n \in N$ , put $P_{n} = (- n - 1, n] \in P$ . This increasing sequence covers $R$ . Since $P_{n} \subseteq [- n - 1, n]$ and both measures are finite on compact sets, $μ (P_{n}) = ν (P_{n}) < \infty$ by the hypothesis.

2.1step 1.1step 1.2L2∎

The generating pi-system from step 1.1 and the finite-measure exhaustion from step 1.2 meet [L2], which gives $μ (E) = ν (E)$ for every Borel $E \subseteq R$ .

Depends on

Used by

Dependency tree · two levels

14 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 1.16 (standard reference, not scraped)

The interval data on (a,b](a,b] determines the Borel measure uniquely