Monitoring programs for finite state properties is challenging due to high memory and execution time overheads it incurs. Some events if skipped or lost naturally can reduce both overheads, but lead to uncertainty about the current monitor state. In this work, we present a theoretical framework to model these lossy event streams and provide a construction for a monitor which observes them without producing false positives. The constructed monitor is optimally sound among all complete monitors. We model several loss types of practical relevance using our framework and provide construction of smaller approximate monitors for properties with a large number of states.