Birkhoff Ergodic Theorem

The Birkhoff Ergodic Theorem states that for certain types of systems that evolve over time in a predictable way, the average value of a measurement taken along a single, long journey through the system will generally be the same as the average taken over all possible states of the system. In essence, if you observe a system long enough, your time-based observations will reflect the system's overall statistical behavior. This theorem provides a rigorous foundation for understanding how long-term averages relate to the inherent properties of complex, dynamic systems.