Fundamental theorems of calculus for Hausdorff measures on the real line (Q1100299)

The paper is concerned with the foundations of Hausdorff measures theory on the line. Let \(m_{\lambda}\) denote the Hausdorff measure on the line corresponding to a function \(\lambda\). For an increasing continuous function F set \[ D^{\lambda}F(x)=\lim_{\epsilon \to 0}\sup_{y:| x-y| <\epsilon}\frac{F(y)-F(x)}{l(y-x)}. \] The author investigates the conditions of the validity of the generalized Newton-Leibniz formula \[ F(y)- F(x)=\int^{y}_{x}(D^{\lambda}F)dm_{\lambda}. \]