Talk:Hausdorff content

There is also a version with the "correct" constant

There is also a version where in place of the terms r_j^d of the sum, one multiplies by the constant C(d) to get C(d) r_j^d, which is the simplest extension of the formula for the actual euclidean volume of the d-dimensional unit ball, where d is a nonnegative integer.

Specifically, C(d) = π^d/2 / 𝛤(d/2 + 1) is the euclidean volume of the unit ball in d-dimensional euclidean space, so the simplest extension of this formula to real (nonnegative) values of d is to use the same formula. In this case, each r_j^d in the summation for Hausdorff d-dimensional content is replaced by

C(d) = π^d/2 / 𝛤(d/2 + 1),

where 𝛤 denotes the gamma function. When d = 2k for an integer k ≥ 0, then 𝛤(d/2 + 1) = k!, so in this case,

C(2k) = π^k / k!.

The value of using this version is that it fits Hausdorff d-dimensional content into the same framework as the analogous content in standard, integral dimensions, where Hausdorff d-dimensional content is equal to Lebesgue measure.