First for measurable $f$. The averages converge a.e. to an exactly invariant function $L$, which is therefore measurable for the $σ$-algebra $ℐ$ of invariant sets. For $A ∈ ℐ$ the averages of $𝟙_A f$ converge to $𝟙_A L$, and the integral of the a.e. limit of the averages of an integrable function equals the integral of that function (dominated convergence after truncation, and Fatou for the remainder); so $∫_A L = ∫_A f$ for every $A ∈ ℐ$, which makes $L$ the conditional expectation $𝔼[f | ℐ]$. Convergence in $L¹$ follows by the same truncation. A general integrable $f$ agrees a.e. with a measurable one, and then so do its averages and its conditional expectation.