Empirical measure
= Empirical measure
{title2=$\nu_N=\frac1N\sum_{j=0}^{N-1}\delta_{x_j}$}
For observations or orbit points $x_0,\ldots,x_{N-1}$, the empirical measure is $\nu_N=N^{-1}\sum_{j=0}^{N-1}\delta_{x_j}$. Integrating a test function gives its sample average. For orbit points $x_j=T^jx$, the difference between $T_*\nu_N$ and $\nu_N$ telescopes to $(\delta_{T^Nx}-\delta_x)/N$, so on a compact <metric space> with continuous $T$, every weak limit is invariant.