Measure-theoretic and topological preparations

Abstract

In this section we collect some prerequisites which are necessary for the development of the theory in the subsequent sections. One of our goals will be to show that under certain assumptions the sequence (V_n) of maximal conditional expected rewards is a solution (w_n) of the optimality equation (OE)

$$\begin{array}{*{20}{c}} {{{w}_{n}}\left( h \right): = \mathop{{\sup }}\limits_{{a \in {{D}_{n}}\left( h \right)}} {\mkern 1mu} \left[ {{{r}_{n}}\left( {h,a} \right) + \int {{{q}_{n}}\left( {h,a,ds} \right){{w}_{{n + 1}}}\left( {h.a.s} \right)} } \right],} \hfill \\ { = :{{U}_{n}}{{w}_{{n + 1}}}\left( h \right),n \in N,h \in {{H}_{n}}} \hfill \\ \end{array} $$

(12.1)

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