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Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization. (English) Zbl 0779.49030
Summary: Let $$(S,{\mathcal A},\mu)$$ denote a positive measure space, where $${\mathcal A}$$ is the Borel $$\sigma$$-algebra of the Polish space $$S$$ and where it is assumed that $$\mu$$ is a bounded measure. The main result concerns $$\text{epi-}\lim_{\nu \to \infty}E^ \nu f=Ef$$ under six different conditions, where $$f:X\times S \to \mathbb{R} \cup\{+\infty\}$$, $$X$$ a Banach space, $$Ef(x)=\int f(x,s)d \mu(s)$$, $$E^ \nu f(x)=\int f(x,s)d \mu^ \nu(s)$$, $$x \in X$$, and where $$\mu^ \nu$$ is a sequence of measures converging weakly to $$\mu$$.

##### MSC:
 49K27 Optimality conditions for problems in abstract spaces 49J45 Methods involving semicontinuity and convergence; relaxation
##### Keywords:
positive measure space; Polish space