In this study, we employ the established Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, along with the telescoping series method, to establish an observability inequality for the degenerate parabolic equation over measurable subsets in the time-space domain. As a direct application, we formulate a captivating Stackelberg–Nash game problem and provide a proof of the existence of its equilibrium. Additionally, we characterize the set of Stackelberg–Nash equilibria and delve into the analysis of a norm optimal control problem.