thoughts of the Banach

With the advent of analytic geometry, however, Euclidean geometry became reinterpreted as the study of Cartesian products ${{\bf R}^d}$ of the real line ${{\bf R}}$. Using this analytic foundation rather than the classical geometrical one, it was no longer intuitively obvious how to define the measure ${m(E)}$ of a general subset ${E}$ of ${{\bf R}^d}$; we will refer to this (somewhat vaguely defined) problem of writing down the