 Tail measurability of limits of sample averages

For a sequence of finite real random variables, ${\mathrm{limsup}}_n{n}^{-1}{\sum }_{j=1}^n{X}_j$ 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.