Orthogonal Measures on the Boundary of a Riemann Surface and Polynomial Hull of Compacts of Finite Length

Abstract

Letμbe an orthogonal measure with compact support of finite length in $\text{C}$ⁿ. We prove, under a very weak hypothesis of regularity on the support (Supp μ) ofμ, that this measure is characterized by its boundary values (in the weak sense of currents) of the current [T]∧ϕ, whereTis an analytic subset of dimension 1 of $\text{C}$ⁿ\Supp μandϕis a holomorphic (1, 0)-form onT. This allows us to prove that the polynomial hullXof a compactumX⊂$\text{C}$ⁿof finite length with a weak regularity assumption is its union with an analytic subset of pure dimension 1 of $\text{C}$ⁿ\X. We also prove that the measureμcan be decomposed into a sum of orthogonal measures will small support. We deduce that a continuous function onXis approximable by polynomials if and only if it islocallyapproximable.