Erdős Problem #232
The statement as posed is true, and a proof exists.
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The proportion \(\lambda(A \cap B_R) / \lambda(B_R)\) of the open ball \(B_R\) of radius `R` centred at the origin that is occupied by `A`. For `R ≤ 0` the ball is empty and the quotient is `0`; this plays no role for the limit \(R \to \infty\).
The upper density \(\overline{\delta}(A) = \limsup_{R \to \infty} \lambda(A \cap B_R) / \lambda(B_R)\), with balls centred at the origin (the upper density does not depend on the centre). The quotients lie in \([0, 1]\), so this is the genuine limsup.
\(m_1 = \sup \overline{\delta}(A)\), where \(A\) ranges over the Lebesgue measurable subsets of \(\mathbb{R}^2\) without two points at distance \(1\). The set of densities is nonempty (take \(A = \emptyset\)) and bounded above by \(1\), so `sSup` is the genuine supremum.
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