The author considers doubling measures that are defined on a proper subset of \({\mathbb R}^n\). More precisely, \(t\)-doubling measures on \(\Omega\), denoted \(\mu\in D_{t}(\Omega)\), are studied and they are defined as positive Borel measures satisfying \[ \mu(B(x,r)\cap \Omega)\leq C\mu(B(x,r/2)\cap\Omega) \] for all balls \(B(x,r)\) for which \(B(x,r/t)\subset\Omega\), when \(t=\infty\) the center of the ball \(B\) should be in \(\Omega\).
The author proves that if \(0<t<t'\leq\infty\), then if \(t'\geq 1\) then \(D_{t'}(\Omega)\setminus D_{t}(\Omega)\) is non-empty for some domain \(\Omega\) which satisfies a quasihyperbolic boundary condition. If \(t'<1\) then \(D_{t'}(\Omega)=D_{t}(\Omega)\) for every proper open set \(\Omega\).
Another result is that if \(\Omega\) satisfies a quasihyperbolic boundary condition and \(\mu\) is \(t\)-doubling for some \(t\) large enough, then the measure of the part of \(\Omega\) lying within a distance \(\varepsilon\) from the boundary \(\partial\Omega\) is dominated by a power of \(\varepsilon\).