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On the measurability of the conjugate and the subdifferential of a normal integrand. (English) Zbl 0854.49014
Consider an integrand \(f : T \times X \to \mathbb{R}\), where \(X\) is a separable Banach space with \(s^*\) (strong topology in \(X^*)\)-separable dual \(X^*\) and \((T, {\mathcal T})\) is an arbitrary measurable space. Because the completeness of a \(\sigma\)-field \({\mathcal T}\) w.r.t. to some measure is difficult to handle the author chooses an approach which does not need the completeness. He shows that also in this case some measurability of an integrand is preserved for the conjugates and the subdifferential multifunction. Here are the main results:
1. If \(t \to \text{epi} f(t,.)\) is an Effros measurable multifunction (i.e., \(f\) is Effros-measurable integrand) then \(x^* \to f^* (t,x^*)\) is \(\text{weak}^*\)-l.s.c. for any \(t \in T\) and \(t \to \text{epi} f^* (t,.)\) is an \(s^*\)-Effros measurable multifunction (i.e., \(w^*\)-l.s.c. normal integrand).
2. The level set multifunction \(t \to \{x^* \in X^* \mid g(t,x^*) \leq b(t)\}\) is \(s^*\)-Effros measurable whenever \(g\) is a \(w^*\)-l.s.c. normal integrand and \(b : T \to \mathbb{R}\) is measurable.
3. The sum of the conjugate functions of two proper Effros-measurable integrands is again a \(w^*\)-l.s.c. normal integrand.
4. Let \(\partial f(t,x)\) be the convex subdifferential of \(f\) w.r.t. \(x\). If \(f\) is a proper normal integrand \((\text{epi} f\) is Effros measurable and \(x \to f(t,x)\) is l.s.c for all \(t \in T)\) and \(u : T \to X\), \(u(t) \in \text{dom} f(t,.)\) for all \(t \in T\) then \(t \to \partial f(t,u(t))\) is Effros measurable.

MSC:
49J52 Nonsmooth analysis
49K27 Optimality conditions for problems in abstract spaces
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
49J45 Methods involving semicontinuity and convergence; relaxation
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