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Let $$𝒦$$ (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that $$\{K\in 𝒦\left(ℝ\right):{\forall }_x\in K(d^{+}(x,K)=1or{d}^{-}(x,K)=1)\}$$ is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.