Descriptive set-theoretical properties of an abstract density operator

Szymon Gła̧b
2009 Open Mathematics  
AbstractLet $$ \mathcal{K} $$(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 $$ \{ K \in \mathcal{K}(\mathbb{R}):\forall _x \in K(d^ + (x,K) = 1ord^ - (x,K) = 1)\} $$ is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.
doi:10.2478/s11533-009-0048-x fatcat:uxusjwn6xfdyhhoglvjhlxg6g4