A Singular Integral Measure for $C^{1,1}$ and $C^1$ Boundaries

The art of analysis involves the subtle combination of approximation, inequalities, and geometric intuition as well as being able to work at different scales. With this subtlety in mind, we present this paper in a manner designed for wide accessibility for both advanced undergraduate students and graduate students. The main results include a singular integral for measuring the level sets of a $C^{1,1}$ function mapping from $\mathbb{R}^n$ to $\mathbb{R}$, that is, one whose derivative is Lipschitz continuous. We extend this to measure embedded submanifolds in $\mathbb{R}^2$ that are merely $C^1$ using the distance function and provide an example showing that the measure does not hold for general rectifiable boundaries.