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Lower semicontinuity and continuity of functions of measures with respect to the strict convergence

Published online by Cambridge University Press:  14 November 2011

S. Delladio
S. Delladio
Affiliation:
Dipartimento di Matematica, Universitá di Trento, 38050 Povo, Italy
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Synopsis

Let Ω be an open subset of Rn. It is well known that, given a suitable real-valued function f on Ω × Rk and a Rk -valued Borel measure µ on Ω, then one can define a real-valued measurefµ on Ω. The object of this note is to define the Ψ-strict convergence of the Rk-valued Borel measures µj to the Rk-valued Borel measure µ, where Ψ: Ω × Rk → [0, + ∞] is a continuous function which is positively homogeneous and convex in the Rk-variable, and to investigate the lower semicontinuity and continuity of the map µ → fμ with respect to the Ψ-strict convergence; here f is positively homogeneous in the Rk-variable and satisfies one suitable convexity condition (related to Ψ).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 265 - 278
Copyright
Copyright © Royal Society of Edinburgh 1991

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