to $g\in L^1(X, \B, m)$. Then $$\lim_{k\to +\infty}\frac{1}{k}\sum_{j=0}^{k-1}g_{k-j}(T^jx)=\E_m(g|\I)(x)$$ almost everywhere and in $L^1$.

Conditional measures {#Sect-con}
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Here we give a brief introduction to Rohlin’s theory of Lebesgue spaces, measurable partitions and conditional measures. The reader is referred to [@Rohlin1949; @Parry1969; @EinsiedlerWard2011] for more details.

A probability space $(X, \B, m)$ is called a [*Lebesgue space*]{} if it is isomorphic to a probability space which is the union of $[0,s]$ ($0\leq s\leq 1$) with Lebesgue measure and a finite or countable number of atoms. Now let $(X, \B,m)$ be a Lebesgue space. A [*measurable partition*]{} $\eta$ of $X$ is a partition of $X$ such that, up to a set of measure zero, the quotient space $X/\eta$ is separated by a countable number of measurable sets $\{B_i\}$. The quotient space $X/\eta$ with its inherited probability space structure, written as $(X_\eta, \B_\eta, m_\eta)$, is again a Lebesgue space. Also, any measurable partition $\eta$ determines a sub-$\sigma$-algebra of $\B$, denoted by $\widehat{\eta}$, whose elements are unions of elements of $\eta$. Conversely, any sub-$\sigma$-algebra $\B'$ of $\B$ is also countably generated, say by $\{B_i'\}$, and therefore all the sets of the form $\cap A_i$, where $A_i=B_i^\prime$ or its complement, form a measurable partition. In 