Another point of view on Kusuoka's measure

Cheeger's energy on the Harmonic Sierpinski Gasket

“…We can afford this arbitrariness since it is standard ( [13], see [3] for an alternative proof) that {a, b, c} is a null set for Kusuoka's measure.…”

“…In this section, we recall the definition and some properties of Kusuoka's measure ( [13]); we follow the approach of [3].…”

“…We say that µ ∈ M(S, M 2 ) is semipositive definite if µ(E) is a semipositive definite matrix for all Borel sets E ⊂ S. Lusin's theorem easily implies ( [3]) that µ is semipositive definite if and only if…”

“…The following proposition is proven as in the scalar case, for which we refer to [16] and [20]; the details for the matrix case are in [3].…”

“…In section 1 below we recall the main definitions and properties of the Harmonic Sierpinski gasket and Cheeger's energy. In section 2 we introduce Kusuoka's measure, following the approach of [3]. In section 3, we study the shape of smaller and smaller cells of the gasket: heuristically, they are "skinny" triangles with the long side aligned with v(x).…”