On a measure-theoretic area formula

Abstract

Abstract. We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathéodory measure. We focus our attention in the case this measure is the spherical Hausdorff measure, giving a metric measure area formula. Our point consists in using certain covering derivatives as "generalized densities". Some consequences for the sub-Riemannian Heisenberg group are also pointed out.

Towards a theory of area in homogeneous groups

“…Observing that F b covers any subset finely, according to the terminology in [18, 2.8.1] and that condition (7.1) holds, we can apply Theorem 11 in [43] to the metric space (G, d), establishing the following result. Theorem 7.2.…”

“…The spherical Federer density θ α (µ, ·) in (7.5) was introduced in [43]. We will use its explicit representation…”

A rough calculus approach to level sets in the Heisenberg group

“…where θ S 2 d (x) is the (upper) spherical Federer density of γ (|ϑ| L 1 I) at x ∈ H. This density, introduced in [21], can be equivalently defined as follows:…”

“…Once we prove that the assumptions in [21,Theorem 11] are satisfied, it will suffice to show that θ S 2 d (x) = 1, for S 2 d -a.e. x ∈ γ(I).…”

$$\Gamma $$ Γ -Convergence of variational functionals with boundary terms in Stein manifolds

“…If X is separable and endowed with a Radon measure µ, absolutely continuous with respect to the m-dimensional spherical Hausdorff measure S m , by [30] (see also [24]), the area formula for µ with respect to S m i.e. for any Borel set B may fail to be true in general, if the m-dimensional density Θ * m F (µ, ·) is replaced by the centered m-dimensional density Θ * m (µ, ·) (see Definition 5.5).…”