LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Stieltjes inversion and uniqueness for probability measures on the real line

Statement

Assume the Axiom of Countable Choice.

Let $μ$ be a Borel probability measure on $R$ , with Stieltjes transform $m_{μ} (z) = \int (x - z)^{- 1} d μ (x)$ for $Im z > 0$ . For every $a < b$ and $x \in R$ , $μ ((a, b)) + \frac{1}{2} μ ({a}) + \frac{1}{2} μ ({b}) = \lim_{η ↓ 0} \frac{1}{π} \int_{a}^{b} Im m_{μ} (u + i η) d u, μ ({x}) = \lim_{η ↓ 0} η Im m_{μ} (x + i η) .$ If two Borel probability measures on $R$ have the same Stieltjes transform on the upper half-plane, then they are equal.

Statement references

Bai and Silverstein, Appendix B.2, printed pp. 514–523 (complete book PDF).

Junyi Liao, §1.3.1, printed pp. 17–18 (complete lecture notes).

Facts & Assumptions

Given: Countable choice, a Borel probability measure $μ$ , and its Stieltjes transform under the $(x - z)^{- 1}$ convention.

[A1]

The Axiom of Countable Choice supplies a choice function for each countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The transform is $m_{μ} (z) = \int (x - z)^{- 1} d μ (x)$ for $Im z > 0$ (Stieltjes transform of a probability measure on the real line).

[F2]

Fubini permits interchanging the integrals for an integrable function on the product measure space (Fubini's theorem for L^1 functions on a sigma-finite product).

[F3]

Dominated convergence passes pointwise limits through an integral under an integrable bound (Dominated convergence).

[F4]

A pi-system generating a sigma-algebra determines any containing lambda-system (Dynkin's pi-lambda theorem).

[F5]

Tonelli's theorem identifies the product integral of a nonnegative measurable function on a sigma-finite product with either iterated integral (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F6]

Under countable choice, the Lebesgue measure $λ$ is a measure and every Borel set is Lebesgue measurable; its restriction to a bounded interval is finite, hence sigma-finite (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ , Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

Proof

Given: Countable choice, $μ$ , and, for the uniqueness claim, two Borel probability measures $μ, ν$ with equal transforms.

1.1A1F1F2F3F5F6givenalgebra

The kernel $K (x, u) = η / ((x - u)^{2} + η^{2})$ is continuous on $R^{2}$ and hence Borel measurable; rational open rectangles form a countable basis, so $B (R^{2}) = B (R) \otimes B (R) \subseteq B (R) \otimes L (R)$ . Restricting the second coordinate to the Lebesgue-measurable interval $[a, b]$ preserves product measurability. Its integral in $u$ is $\int_{a}^{b} \frac{η}{(x - u)^{2} + η^{2}} d u = \arctan \frac{b - x}{η} - \arctan \frac{a - x}{η} \leq π .$ By [F6], Lebesgue measure restricted to $[a, b]$ is finite, and the probability measure $μ$ is finite. Tonelli [F5] shows that the product integrand has integral at most $π μ (R) = π$ , so it belongs to $L^{1}$ on the product space. Fubini [F2] therefore gives $\frac{1}{π} \int_{a}^{b} Im m_{μ} (u + i η) d u = \int K_{η, a, b} (x) d μ (x), K_{η, a, b} (x) = \frac{1}{π} (\arctan \frac{b - x}{η} - \arctan \frac{a - x}{η}) .$ The kernel lies in $[0, 1]$ and tends pointwise to $1$ on $(a, b)$ , $1 / 2$ at either endpoint, and $0$ elsewhere. [F3] gives the interval formula in the Statement, including endpoint atoms.

1.2F1F3algebra

For any $x \in R$ , $η Im m_{μ} (x + i η) = \int \frac{η^{2}}{(t - x)^{2} + η^{2}} d μ (t) .$ The integrand is bounded by $1$ and tends to $1_{{x}} (t)$ as $η ↓ 0$ . Dominated convergence [F3] proves the atom formula.

2.1step 1.1step 1.2algebra

If $m_{μ} = m_{ν}$ , step 1.2 gives equality of their point masses at every $x$ . Step 1.1 then gives $μ ((a, b)) = ν ((a, b))$ for every $a < b$ .

3.1F4step 2.1algebra∎

The family $P = {\emptyset, R} \cup {(a, b) : a < b}$ is a pi-system: the intersection of two open intervals is another open interval or empty. It generates the Borel sigma-algebra because open intervals form a basis. The class of Borel sets on which $μ$ and $ν$ agree is a lambda-system: it contains $R$ because both measures are probabilities, is closed under complements by finite total mass, and is closed under disjoint countable unions by countable additivity. Step 2.1 and [F4] therefore give equality on every Borel set, so $μ = ν$ .

Depends on

Used by

Dependency tree · two levels

50 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

Bai and Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed., Appendix B.2 (standard reference, not scraped)
Junyi Liao, Random Matrix Theory, §1.3.1 (standard reference, not scraped)