Markov Decision Processes (MDP) with explicit measurement cost are a class of en- vironments in which the agent learns to maximize the costed return. Here, we define the costed return as the discounted sum of rewards minus the sum of the explicit cost of measuring the next state. The RL agent can freely explore the relationship between actions and rewards but is charged each time it measures the next state. Thus, an op- timal agent must learn a policy without making a large number of measurements. We propose the active measure RL framework (Amrl) as a solution to this novel class of problem, and contrast it with standard reinforcement learning under full observability and planning under partially observability. We demonstrate that Amrl-Q agents learn to shift from a reliance on costly measurements to exploiting a learned transition model in order to reduce the number of real-world measurements and achieve a higher costed return. Our results demonstrate the superiority of Amrl-Q over standard RL methods, Q-learning and Dyna-Q, and POMCP for planning under a POMDP in environments with explicit measurement costs.