A definition of fractional k-dimensional measure: bridging the gap between fractional length and fractional area

Here we introduce a fractional notion of $k$-dimensional measure, $0\leq k<n$, that depends on a parameter $\sigma$ that lies between $0$ and $1$. When $k=n-1$ this coincides with the fractional notions of area and perimeter, and when $k=1$ this coincides with the fractional notion of length. It is shown that, when multiplied by the factor $1-\sigma$, this $\sigma$-measure converges to the $k$-dimensional Hausdorff measure up to a multiplicative constant that is computed exactly. We also mention several future directions of research that could be pursued using the fractional measure introduced.