= Tail measurability of limits of sample averages
Let $(X_j)$ be a <sequence> of finite real <random variables>, and write $M_n=n^{-1}\sum_{j=1}^nX_j$ for its <sample mean>. For fixed $m$ and $n>m$, put $M_n^{(m)}=n^{-1}\sum_{j=m+1}^nX_j$. Then, at each outcome,
$$
M_n-M_n^{(m)}=\frac{X_1+\cdots+X_m}{n}\longrightarrow0.
$$
Consequently $\limsup_n M_n=\limsup_n M_n^{(m)}$. The right-hand <limit superior> is a <measurable function> of $X_{m+1},X_{m+2},\ldots$ for every $m$, so the extended-real <random variable> $\limsup_n M_n$ is measurable with respect to the <tail sigma-algebra>. The same holds for the <limit inferior>. Thus the <event> of convergence to a finite <limit of a sequence>, and the <event> $\{M_n\to a\}$ for any specified real $a$, are <tail events>.
For a worked deterministic example, let $x_j$ alternate $0,1,0,1,\ldots$, and replace its first three entries $0,1,0$ by $1,0,1$. For every $n\geq3$ the two <sample means> differ by exactly $1/n$, so both converge to $1/2$. Finite edits can change small-sample answers while preserving the infinite <limit of a sequence>.
Under <independence>, <Kolmogorov's zero–one law> makes each specified convergence <event> have <probability> zero or one. It does not select which value occurs. If a finite <random variable> measurable with respect to the <tail sigma-algebra> $L$ exists, the <events> $\{L\leq q\}$ for rational $q$ each have <probability> zero or one; <continuity of probability> and their nesting force $L$ to be constant <almost surely>. For fair <independent> <Bernoulli trials>, the <strong law of large numbers> establishes $M_n\to1/2$ <almost surely>; the zero–one conclusion alone proves neither convergence nor the value $1/2$.
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