Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization. (English) Zbl 0779.49030

Lucchetti, Roberto; Wets, R. J.-B.

Stat. Decis. 11, No. 1, 69-84 (1993).

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