We study content fetching and delivery in an edge caching system consisting of a sensor, a set of users, and an aggregator. The users occasionally request the aggregator for the dynamically varying sensor measurements also called the content. The aggregator may fetch the fresh content from the sensor and serve, may serve the locally cached versions, or may not serve at all. Content fetching and delivery incur fetching and transmission costs, respectively, and serving stale content also incurs age cost. We study optimal content fetching and delivery problem, aiming to minimize the time-averaged content fetching, transmission, and age costs. The problem lends itself to Markov decision problem framework albeit with a high dimensional state space. We first consider a single-user problem and derive the optimal policy. We then propose a heuristic for the multi-user problem, which is obtained by combining solutions to certain fictitious versions of the single-user problems. We numerically illustrate the derived content fetching and delivery policies and their properties.