Graphs of Functions, Regular Sets and S-Straight Sets

Abstract

A subset $E$ of $\Bbb{R}^p$ is s-straight if $E$ has finite Hausdorff s-dimensional outer measure which equals its Method I s-outer measure. The graph of a continuously differentiable function is shown to be the countable union of closed 1-straight sets together with a set of Hausdorff 1-measure zero. This result is extended to the graphs of absolutely continuous functions and to regular sets.