Strategies for selecting base probability measures that both respect market valuations of traded cash flows and simultaneously monitor departures from physical measures are suggested, proposed and implemented in an illustrative option hedging context. Mathematical objectives for financial decision making that are financial are advocated and developed using the theory of nonlinear expectations. Financial objectives should equate the additional value of a dollar to a dollar. More exactly, the value of a claim paying out an additional dollar, should exceed the original claim value by a dollar.