We introduce a novel framework for analyzing reinforcement learning (RL) in continuous state-action spaces, and use it to prove fast rates of convergence in both off-line and on-line settings. Our analysis highlights two key stability properties, relating to how changes in value functions and/or policies affect the Bellman operator and occupation measures. We argue that these properties are satisfied in many continuous state-action Markov decision processes. Our analysis also offers fresh perspectives on the roles of pessimism and optimism in off-line and on-line RL.