Stochastic optimal control control problems with merely measurable coefficients are not well understood. In this manuscript, we consider fully non-linear stochastic optimal control problems in infinite horizon with measurable coefficients and (local) uniformly elliptic diffusion. Using the theory of $L^p$-viscosity solutions, we show existence of an $L^p$-viscosity solution $v\in W_{\rm loc}^{2,p}$ of the Hamilton-Jacobi-Bellman (HJB) equation, which, in turn, is also a strong solution (i.e. it satisfies the HJB equation pointwise a.e.). We are then led to prove verification theorems, providing necessary and sufficient conditions for optimality. These results allow us to construct optimal feedback controls and to characterize the value function as the unique $L^p$-viscosity solution of the HJB equation. To the best of our knowledge, these are the first results for fully non-linear stochastic optimal control problems with measurable coefficients. We use the theory developed to solve a stochastic optimal control problem arising in economics within the context of optimal advertising.