Triangular averages of finite-past information converge to the infinite-past average

For each $K$, put $U_K={\sup }_{j\ge K}\mathrm{\mid }{g}_j-g\mathrm{\mid }$. Then $U_K\downarrow 0$ $\mu$-almost everywhere and $0\le U_K\le 2G$ almost everywhere, since $g\le G$ as a pointwise limit. Dominated convergence [F4] gives $\int U_{K}d\mu \to 0$.

Since $g_i\to g$ almost everywhere and $\mathrm{\mid }{g}_i-g\mathrm{\mid }\le 2G$, [F4] gives $\int \mathrm{\mid }{g}_i-g\mathrm{\mid }d\mu \to 0$. By [F3, F5], $\int \mathrm{\mid }{g}_i-g\mathrm{\mid }\circ T^{i}d\mu =\int \mathrm{\mid }{g}_i-g\mathrm{\mid }d\mu$, so linearity [F7] makes the integral of the absolute triangular error $n^{-1}{\sum }_{i<n}\int \mathrm{\mid }{g}_i-g\mathrm{\mid }d\mu \to 0$ by Cesaro averaging. Thus the error tends to zero in $L^1(\mu )$.

Fix $K$. For $i\ge K$, $\mathrm{\mid }{g}_i-g\mathrm{\mid }\le U_K$. Hence for $n>K$, the average of $\mathrm{\mid }{g}_i-g\mathrm{\mid }\circ T^i$ is bounded by the first $K$ terms divided by $n$, plus $n^{-1}{\sum }_{i=K}^{n-1}{U}_K\circ T^i$. The first part tends to zero almost everywhere. By [F1], the second part converges almost everywhere to the Birkhoff limit $U_K^{\ast }$; by [F2], the averages converge to $U_K^{\ast }$ in $L^1(\mu )$. By [F3, F5, F7], every average has integral $\int U_{K}d\mu$. Thus [F6] and the $L^1$ convergence give $\mid \int U_{K}d\mu -\int U_K^{\ast }d\mu \mid \le \mathrm{\parallel }{A}_n{U}_K-U_K^{\ast }{\mathrm{\parallel }}_1\to 0$. Therefore $U_K^{\ast }\in L^1(\mu )$ and $\int U_K^{\ast }d\mu =\int U_{K}d\mu$.

Apply step 2.1 to every integer $K$ on the intersection of the resulting countably many full-measure sets. The limsup of the left side is at most $U_K^{\ast }$ for each $K$. Since $U_K$ decreases pointwise, the averages and their limits $U_K^{\ast }$ decrease almost everywhere. They are nonnegative and bounded by the integrable $U_1^{\ast }$; dominated convergence [F4] gives $\int {\inf }_K{U}_K^{\ast }d\mu ={\lim }_K\int U_K^{\ast }d\mu ={\lim }_K\int U_{K}d\mu =0$. Thus ${\inf }_K{U}_K^{\ast }=0$ almost everywhere, and the limsup is zero. Therefore $n^{-1}{\sum }_{i<n}\mathrm{\mid }{g}_i-g\mathrm{\mid }(T^i\omega )\to 0$ almost everywhere.

By [F1]–[F2], the Birkhoff averages of $g$ converge almost everywhere and in $L^1(\mu )$ to their Birkhoff limit. Steps 3.1 and 1.2 show the triangular averages differ from these averages by quantities tending to zero almost everywhere and in $L^1(\mu )$, respectively. Hence both modes of convergence hold with the same limit.