Ergodic time-series sample means converge almost surely and in L1

Define $Y_k=X_{k+1}$ for $k\ge 0$. For any finite indices $0\le k_1<\cdots <k_r$ and shift $m\ge 0$, [F1] applied to the times $k_1+1,\dots ,k_r+1$ gives $\mathcal{L}(Y_{k_1+m},\dots ,Y_{k_r+m})=\mathcal{L}(Y_{k_1},\dots ,Y_{k_r})$. Thus $Y$ is strictly stationary in the one-sided convention of [F2]. Taking $r=1$ shows $Y_0=X_1$ has the same law as $X_0$, so it is integrable; the canonical shift is ergodic by the given assumption.

By [F3] the assumed ergodicity is exactly ergodicity of the canonical path shift required by [F4]. AC [A1], integrability from step 1.1, strict stationarity from step 1.1, and that ergodic shift satisfy the hypotheses of [F4]. Hence $n^{-1}{\sum }_{k=0}^{n-1}{Y}_k$ converges almost surely and in $L^1$ to $\mathbb{E}[Y_0]=\mathbb{E}[X_0]$. Since this average is exactly $n^{-1}{\sum }_{t=1}^n{X}_t={\bar{X}}_n$, the asserted convergence follows. The choice assumption is inherited through [F4]'s conditional-expectation identification; this proof adds no other choice use.