ergodic measures

An ergodic measure describes a way to analyze dynamic systems by looking at their long-term behavior. Imagine observing a process over a long time; if the time you spend in each region of the system reflects its overall statistical properties, the system is ergodic. Conversely, an ergodic measure means that the system, over time, explores all parts in a representative way, so time averages (what you observe over a long period) match space averages (what is true across the entire system). It ensures that individual long-term observations are meaningful for understanding the whole system's behavior.