Calculus of variations and descriptive set theory

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)