Integral geometric measure in separable Banach space

Abstract

We define an \(m\) dimensional integral geometric measure in separable Banach spaces. In case \(X\) is a Banach space with the Radon-Nikodým, the integral geometric measure of an \(m\) rectifiable subset \(M\subseteq X\) is bounded below by its Hausdorff measure. Moreover, we give an explicit formula for the computation of the integral geometric measure. We also define a corresponding integral geometric mass of \(m\) dimensional rectifiable chains in \(X\) with coefficients in a complete Abelian group and apply our results to establish it is lower semicontinuous with respect to flat norm convergence.