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Here we introduce a fractional notion of $k$-dimensional measure, $0\leqk<n$, 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, andwhen $k=1$ this coincides with the fractional notion of length. It is shownthat, when multiplied by the factor $1-\sigma$, this $\sigma$-measure convergesto the $k$-dimensional Hausdorff measure up to a multiplicative constant thatis computed exactly. We also mention several future directions of research thatcould be pursued using the fractional measure introduced.

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