Run-time monitoring can provide important insights about a program's behavior and, for simple properties, it can be done efficiently. Monitoring properties describing sequences of program states and events, however, can result in significant run-time overhead. In this paper we present a novel approach to reducing the cost of run-time monitoring of path properties. Properties are composed to form a single integrated property that is then systematically decomposed into a set of properties that encode necessary conditions for property violations. The resulting set of properties forms a lattice whose structure is exploited to select a sample of properties that can lower monitoring cost, while preserving violation detection power relative to the original properties. Preliminary studies for a widely used Java API reveal that our approach produces a rich, structured set of properties that enables control of monitoring overhead, while detecting more violations than alternative techniques.