A product of doubling measure on the real line (Q2455362)

The author studies (positive real-valued \(\sigma\)-additive Borel) measures on the real line. A measure \(\mu\) on \(\mathbb{R}\) is called doubling if there exists a constant \(K\geq 1\) such that \(\mu([a-2t,a+2t])\leq K\mu([a-t,a+t])\) for all \(a,t\in\mathbb{R}\) with \(t>0\). The author defines the product \(\mu\bullet\nu\) of two doubling measures \(\mu\) and \(\nu\) as the measure satisfying \((\mu\bullet\nu)([0,a])=\mu([0,a])\cdot \nu([0,a])\) and \((\mu\bullet\nu)([a,0])=\mu([a,0])\cdot \nu([a,0])\) for all \(a>0\). It is shown that then \(\mu\bullet\nu\) is doubling, too. It follows that doubling measures on \(\mathbb{R}\) (with this product and the usual sum) form a semiring.