The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry
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We provide a precise characterization of the weak sequential closure of nonempty, closed, decomposable sets in Lebesgue spaces. Therefore, we have to distinguish between the purely atomic and the nonatomic regime. In the latter case, we get a convexification effect which is related to Lyapunov’s convexity theorem, and in the former case, the weak sequential closure equals the strong closure. The characterization of the weak sequential closure is utilized to compute the limiting normal cone to nonempty, closed, decomposable sets in Lebesgue spaces. Finally, we give an example for the possible nonclosedness of the limiting normal cone in this setting.
KeywordsDecomposable set Lebesgue spaces Limiting normal cone Measurability Weak sequential closure
Mathematics Subject Classification (2010)49J53 28B05 90C30
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