Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure

TL;DR

This paper introduces an alternative geometric measure to Hausdorff measure, emphasizing integration and providing simpler tools for geometric analysis.

Contribution

It constructs a new measure that satisfies the Area Formula and offers a more accessible approach for specific applications in geometric analysis.

Findings

The measure serves as an alternative to Hausdorff measure.

It satisfies the Area Formula and is minimal with this property.

Provides a simpler introduction to geometric measure theory.

Abstract

We employ methods of geometry and generalized convergence to construct a geometric measure that serves as an alternative to the integer-dimension Hausdorff measure. This construction prioritizes integration, yields the Area Formula as a byproduct of the construction and the Coarea Formula follows indirectly from the smooth case. Furthermore, this is the smallest measure that satisfies the Area Formula. Though this construction is not as general as Hausdorff measure, it provides a much simpler introduction to the topic and is enough for certain applications to Geometric Analysis.