Tail measurability of limits of sample averages
= Tail measurability of limits of sample averages
For a sequence of finite real <random variables>, $\limsup_n n^{-1}\sum_{j=1}^nX_j$ is measurable with respect to its <tail sigma-algebra>: deleting finitely many summands changes each <sample mean> by a quantity tending to zero. When the <random variables> are <independent random variables>, the <Kolmogorov zero-one law> makes every finite such limit constant <almost surely>.