TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-10-02

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.

Existence and uniqueness of the equilibrium measure

Statement

Assume the Axiom of Choice. Let $K \subseteq C$ be nonempty and compact with $cap (K) > 0$ . Then there is exactly one Borel probability measure $μ_{K}$ on $K$ with

$I (μ_{K}) = V_{K} = \inf_{μ \in P (K)} I (μ) < + \infty .$

The measure $μ_{K}$ is the equilibrium measure of $K$ . If $cap (K) = 0$ then $V_{K} = + \infty$ , every $μ \in P (K)$ has $I (μ) = + \infty$ , and no equilibrium measure is asserted.

The Axiom of Choice is spent twice: through Countable Choice for the minimizing sequence, and through the weak sequential compactness of probability laws of Probability laws on a compact metric space have weakly convergent subsequences. The energy lower semicontinuity used below is choice-free (Lower semicontinuity of logarithmic potential and energy).

Facts & Assumptions

Given: a nonempty compact set $K \subseteq C$ with $cap (K) > 0$ , the probability measures $P (K)$ on $K$ , the Robin constant $V_{K}$ and the logarithmic energy $I$ of Robin constant and logarithmic capacity of a compact set and Logarithmic potential and energy of a positive compactly supported measure, and the Axiom of Choice (The Axiom of Choice).

[F1]

$P (K)$ is the set of Borel probability measures on $K$ (Probability measures and probability spaces), each of compact support contained in $K$ ; $V_{K} = \inf_{μ \in P (K)} I (μ) \in (- \infty, + \infty]$ and $cap (K) = \exp (- V_{K})$ when $V_{K} < + \infty$ and $cap (K) = 0$ when $V_{K} = + \infty$ , so $cap (K) > 0$ is equivalent to $V_{K} < + \infty$ (Robin constant and logarithmic capacity of a compact set).

[F2]

For finite positive Borel measures $μ, ν$ of compact support the mixed energy $I (μ, ν) = \iint k d μ d ν \in (- \infty, + \infty]$ is symmetric, $I (μ, ν) = I (ν, μ)$ , and it is computed from the shifted nonnegative kernel $k_{R} = k + \log R$ with $R > diam (supp μ \cup supp ν)$ by $I (μ, ν) = \iint k_{R} d μ d ν - μ (C) ν (C) \log R$ (Logarithmic potential and energy of a positive compactly supported measure).

[F3]

If $μ_{n}, μ \in P (K)$ with $μ_{n} \Rightarrow μ$ in the sense of Weak convergence of borel probability measures, then $U^{μ} (z) \leq {lim inf}_{n} U^{μ_{n}} (z)$ for every $z \in C$ and $I (μ) \leq {lim inf}_{n} I (μ_{n})$ (Lower semicontinuity of logarithmic potential and energy).

[F4]

Assume the Axiom of Choice: every sequence in $P (K)$ has a subsequence converging weakly to some element of $P (K)$ (Probability laws on a compact metric space have weakly convergent subsequences).

[F5]

Assume Countable Choice, and let $μ, ν$ be finite positive Borel measures on $C$ with compact support, equal total mass and finite energy. Then $I (μ, ν)$ is finite, $I (μ - ν) : = I (μ) - 2 I (μ, ν) + I (ν)$ is a real number, $I (μ - ν) \geq 0$ , and $I (μ - ν) = 0$ if and only if $μ = ν$ (Strict positivity of logarithmic energy for a zero-mass signed charge).

[F6]

The Axiom of Choice implies Dependent Choice, which implies Countable Choice (AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F7]

If $S \subseteq R$ is nonempty and bounded below with infimum $L$ , then for every $ε > 0$ there is $s \in S$ with $s < L + ε$ (Epsilon characterisation of the infimum).

[F8]

Finite nonnegative weighted sums of measures are measures, and a convex combination $(1 - t) μ + t ν$ with $0 \leq t \leq 1$ of probability measures is again a probability measure (Nonnegative scalar multiples and countable weighted sums of measures are measures, Probability measures and probability spaces).

Proof

technique · direct

1.1F1F7given

Since $cap (K) > 0$ , [F1] and the finite-energy construction in Robin constant and logarithmic capacity of a compact set give the nonempty real set $E_{K} : = {I (μ) : μ \in P (K), I (μ) < + \infty}$ with $\inf E_{K} = V_{K} \in R$ . It is bounded below by $- \log diam K$ . The infinite energies do not alter its real lower bounds. Applying [F7] to $E_{K}$ , for each $n \geq 1$ there is a finite energy below $V_{K} + 1 / n$ , so ${μ \in P (K) : I (μ) < V_{K} + 1 / n}$ is nonempty.

2.1step 1.1F6

By [F6] the Axiom of Choice yields Countable Choice, so there is a sequence $(μ_{n})_{n \geq 1}$ in $P (K)$ with $I (μ_{n}) < V_{K} + 1 / n$ for every $n$ .

3.1step 2.1F4

By [F4], which assumes the Axiom of Choice of the hypothesis, the sequence $(μ_{n})$ has a subsequence $(μ_{n_{k}})_{k \geq 1}$ and a limit $μ \in P (K)$ with $μ_{n_{k}} \Rightarrow μ$ .

4.1step 3.1F3F1

Applying [F3] to the weakly convergent subsequence of step 3.1 and using step 2.1 along it gives $I (μ) \leq {lim inf}_{k} I (μ_{n_{k}}) \leq {lim inf}_{k} (V_{K} + 1 / n_{k}) = V_{K}$ , while $V_{K} \leq I (μ)$ holds because $V_{K}$ is an infimum over $P (K) ∋ μ$ ; hence $I (μ) = V_{K} < + \infty$ .

5.1step 4.1F2F5F8given

For uniqueness let $μ, ν \in P (K)$ satisfy $I (μ) = I (ν) = V_{K}$ ; both have compact support in $K$ , finite energy and total mass $1$ , so [F5] applies to the pair and the average $σ : = \frac{1}{2} μ + \frac{1}{2} ν$ lies in $P (K)$ by [F8], whence $I (σ) \geq V_{K}$ . Expanding the double integral of $σ \otimes σ = \frac{1}{4} μ \otimes μ + \frac{1}{4} μ \otimes ν + \frac{1}{4} ν \otimes μ + \frac{1}{4} ν \otimes ν$ and using $I (μ, ν) = I (ν, μ)$ from [F2] gives the finite value $I (σ) = \frac{1}{4} I (μ) + \frac{1}{2} I (μ, ν) + \frac{1}{4} I (ν)$ ; comparing with the companion expansion $I (μ - ν) = I (μ) - 2 I (μ, ν) + I (ν)$ of the same bilinear form from [F5], this says $I (\frac{1}{2} (μ + ν)) = \frac{1}{2} I (μ) + \frac{1}{2} I (ν) - \frac{1}{4} I (μ - ν)$ . Substituting $I (μ) = I (ν) = V_{K}$ yields $V_{K} \leq \frac{1}{2} V_{K} + \frac{1}{2} I (μ, ν)$ , that is, $I (μ, ν) \geq V_{K}$ .

6.1step 5.1F5

With $σ$ as in step 5.1 the signed measure $μ - ν$ also meets the hypotheses of [F5], so $I (μ - ν) = I (μ) - 2 I (μ, ν) + I (ν) \leq V_{K} - 2 V_{K} + V_{K} = 0$ by step 5.1, while [F5] gives $I (μ - ν) \geq 0$ ; hence $I (μ - ν) = 0$ and [F5] gives $μ - ν = 0$ , that is, $μ = ν$ , so the minimizer of step 4.1 is the only minimizer and is the stated equilibrium measure $μ_{K}$ .

7.1step 4.1step 6.1F1∎

The zero-capacity case is [F1] verbatim: if $cap (K) = 0$ then $V_{K} = + \infty$ , so an element $μ \in P (K)$ with $I (μ) = V_{K}$ would have $I (μ) = + \infty$ by definition of the extended infimum, and the statement asserts nothing about the existence of such a $μ$ , which completes the proof.

Remarks

The equilibrium measure is a probability on the conductor. The minimizer $μ_{K}$ of the theorem is carried by $K$ , since $P (K)$ consists of the probability measures on $K$ ; this is used by every later item that integrates against $μ_{K}$ over $K$ .

Uniqueness is strict convexity of the energy. The proof shows more than the statement needs: any two finite-energy probabilities of equal mass on a common compact carrier satisfy $I (\frac{1}{2} (μ + ν)) = \frac{1}{4} I (μ) + \frac{1}{2} I (μ, ν) + \frac{1}{4} I (ν)$ and $μ \neq ν$ forces $I (μ - ν) > 0$ by Strict positivity of logarithmic energy for a zero-mass signed charge.

Where the two choice uses sit. Countable Choice selects one measure per level $n$ in step 2.1; the Axiom of Choice itself is the hypothesis of the weak-compactness statement [F4] used in step 3.1. The lower semicontinuity [F3] and the infimum characterization [F7] are choice-free.

Depends on

Used by

Dependency tree · two levels

54 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

E. B. Saff, Logarithmic Potential Theory with Applications to Approximation Theory, §1 (standard reference, not scraped)
B. Khoruzhenko, LTCC Potential Theory notes, §3 (standard reference, not scraped)
C. Kuehn, Introduction to Potential Theory via Applications, §2.3 (standard reference, not scraped)