In this paper, we study the approximate controllability for the stochastic
heat equation over measurable sets, and the optimal actuator location of the
minimum norm controls. We formulate a relaxed optimization problem for both
actuator location and its corresponding minimum norm control into a two-person
zero sum game problem and develop a sufficient and necessary condition for the
optimal solution via Nash equilibrium. At last, we prove that the relaxed
optimal solution is an optimal actuator location for the classical problem