Let \(\mu\) be a locally finite Borel measure on \(R\) (\(R\) — the real line) and let \(b>0\). The measure \(\mu\) is said to be \(b\)-concentrated at the point \(x\) if \(x\in\text{supp}(\mu)\) and \[ \limsup_{h\to 0+}\mu((x- bh,x+bh))\mu((x-h,x+h))^{-1}<b. \] Let \(C_ b(\mu)\) be the set of all points at which \(\mu\) is \(b\)-concentrated. The authors prove that for \(b\leq 1\), \(\mu(C_ b(\mu))=0\) holds for every measure \(\mu\). If \(b>1\) then \(C_ b(\mu)\) is a \(\sigma\)-porous set for every \(\mu\). The Cantor measure (supported by the Cantor ternary set) is investigated in details. It is shown that it is \(b\)-concentrated for every \(b\geq 81\) and it is not \(b\)-concentrated for \(b=4\). The paper contains an application of reached results to generalized Riemann derivative of the distribution function \(f\) of \(\mu\) \((f(x)=\mu([0,x))\) for \(x\geq 0\), \(f(x)=-\mu([x,0))\) for \(x<0)\).