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Calculus of variations and descriptive set theory
Author(s) -
Sofronidis Nikolaos E.
Publication year - 2009
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200810003
Subject(s) - mathematics , space (punctuation) , polish space , locally compact space , borel set , compact space , convergence (economics) , combinatorics , discrete mathematics , calculus (dental) , pure mathematics , mathematical analysis , computer science , medicine , dentistry , separable space , operating system , economics , economic growth
If X is a locally compact Polish space, then LSC( X , ℝ) denotes the compact Polish space of lower semi‐continuous real‐valued functions on X equipped with the topology of epi‐convergence. Our purpose in this article is to prove the following: if –∞ < α < β < ∞ and –∞ < a < b < ∞, while r ∈ ℕ \ {0}, then the set CV of all f ∈ LSC([ α , β ] × [ a , b ] × ℝ, ℝ) for which there is u ∈ C r ([ α , β ], [ a , b ]) such that for any v ∈ C r ([ α , β ], [ a , b ]) we have that ∫ α β f ( x , u ( x ), v ′( x ))d x ≥ ∫ α β f ( x , v ( x ), v ( x ))d x is not Borel (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)