We have a covariance stationary time series. We must assume that the time series was produced by an ergodic process if we are to make the bridge between the realization of the time series that we observe and the population that might have generated it. The assumption of an ergodic process amounts to assuming that the mean of our time series converges in probability to the population mean.
The Law of Large Numbers for a covariance stationary process doesn't seem to add anything to the assumption of ergodicity we've made. Or does it?
(1) What does the Law of Large Numbers for a covariance stationary process give us that we did not already know, having assumed ergodicity? (2) Ergodicity is something untestable about the process that created the time series. The process is either ergodic or not, and we can't know it. Can we think of the LLN for a process as a statement about conditions under which LLN holds, hence a statement about conditions when we can expect a process to be ergodic?