Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 947-972

RECOVERING MEASURES FROM APPROXIMATE VALUES ON BALLS

Blanche Buet and Gian Paolo Leonardi

Université Paris Sud, Département de Mathématiques d'Orsay
91405 Orsay Cedex, France; blanche.buet 'at' math.u-psud.fr

Università di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche
41125 Modena, Italy; gianpaolo.leonardi 'at' unimore.it

Abstract. In a metric space (X,d) we reconstruct an approximation of a Borel measure μ starting from a premeasure q defined on the collection of closed balls, and such that q approximates the values of μ on these balls. More precisely, under a geometric assumption on the distance ensuring a Besicovitch covering property, and provided that there exists a Borel measure on X satisfying an asymptotic doubling-type condition, we show that a suitable packing construction produces a measure ˆμ^q which is equivalent to μ. Moreover we show the stability of this process with respect to the accuracy of the initial approximation. We also investigate the case of signed measures.

2010 Mathematics Subject Classification: Primary 28C15, 28A78, 49Q15.

Key words: Packing measure, premeasure, Besicovitch covering lemma, doubling-type condition.

Reference to this article: B. Buet and G.P. Leonardi: Recovering measures from approximate values on balls. Ann. Acad. Sci. Fenn. Math. 41 (2016), 947-972.

Full document as PDF file

doi:10.5186/aasfm.2016.4160