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Functions and operators on the space of bounded measures. (English) Zbl 0659.46028

Let T be a locally compact metrizable \(\sigma\)-compact space equipped with a positive Radon measure \(\mu\) (in application T can be an open subset of \({\mathbb{R}}^ n\) and \(\mu\) the Lebesgue measure). Let \(f:T\times {\mathbb{R}}^ d\to]-\infty,\infty]\) be a convex normal integrand (that is a measurable function such that \(\forall t\), f(t,\(\cdot)\) is convex lower semi-continuous, shortly l.s.c.). Consider the function (usually called integral functional) \(I_ f\) on \(L^ 1(T,\mu:{\mathbb{R}}^ d\) (shortly \(L^ 1)\) defined by \[ I_ f(u)=\int_{T}f(t,u(t))\mu (dt). \] Denote by \({\mathcal C}_ 0(T;{\mathbb{R}}^ d)\) (shortly \({\mathcal C})\) the space of \({\mathbb{R}}^ d\) valued functions which tend to 0 at infinity. A first question is: what is the \(\sigma (L^ 1,{\mathcal C})\) l.s.c. hull \(\bar I_ f\), of \(I_ f\), when is \(I_ f\) \(\sigma (L^ 1,{\mathcal C})\) l.s.c.? A motivation for the last question arises from optimization.
Consider now the space \({\mathcal M}(T;{\mathbb{R}}^ d)\) (shortly \({\mathcal M})\) of \({\mathbb{R}}^ d\) valued measures (they are set functions defined on the Borel tribe of T). Put \[ F(\lambda)=I_ f(\frac{d\lambda}{d\mu})\quad if\quad \lambda \ll \mu \quad and\quad =\pm \infty \quad otherwise. \] The second question is: what is the \(\sigma\) (\({\mathcal M},{\mathcal C})\) l.s.c. hull \(\bar F\) of F? This question arises from plasticity.
In § 2 we set hypotheses, in § 3 the descriptions of \(\bar F\) and \(\bar I_ f\) are given, as well as a necessary and sufficient condition for \(I_ f\) to be \(\sigma (L^ 1,{\mathcal C})\) l.s.c. and examples. In § 4 we sketch the proofs [all details are in Sem. Anal. Convex, Montpellier 1986, exp. 3, 31 pages] and we give some intermediate results and prepare the way to § 5 where the operator G is defined and some of its properties are stated.

MSC:

46E27 Spaces of measures
46G05 Derivatives of functions in infinite-dimensional spaces
49J52 Nonsmooth analysis
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