= Triangular ergodic averaging lemma
Suppose $f_k\to f$ <almost everywhere> and $\sup_k|f_k|\in L^1$ in a probability <measure-preserving system>. Then
$$
\frac1n\sum_{j=0}^{n-1}f_{n-1-j}(T^jx)
\longrightarrow\mathbb E[f\mid\mathcal I]
$$
<almost everywhere> and in $L^1$, where $\mathcal I$ is the <invariant sigma-algebra>. For the almost-everywhere assertion, bound the terms with index at least $M$ by $\sup_{k\geq M}|f_k-f|$, apply the <Birkhoff ergodic theorem>, and then let $M\to\infty$. The finitely many remaining end terms vanish by the <linear growth bound for integrable observables>.
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