invariant measures

Invariant measures are mathematical tools used to describe the long-term behavior of a system that evolves over time, like a weather pattern or a population. Think of them as a way to assign probabilities to different states so that, when the system runs for a long period, these probabilities stay the same—they're "invariant." This means the measure captures the steady-state distribution of where the system spends its time, helping us understand its typical behavior without being affected by initial starting points. Essentially, invariant measures reveal the system's stable, recurring patterns over the long run.