Let $(X,\mathcal B,\mu,T)$ be an ergodic measure-preserving system, let $\mathcal G\subset\mathcal B$ be a factor $\sigma$-algebra with $T^{-1}\mathcal G=\mathcal G$ modulo $\mu$-null sets, and let $\mathcal P$ be a finite measurable partition. Here $\mathcal G$ is not the invariant $\sigma$-algebra $\mathcal I_T$; it represents information coming from a factor system. For $n\in\mathbb N$, define the finite join

\begin{align*} \mathcal P_{[0,n-1]}:=\bigvee_{k=0}^{n-1}T^{-k}\mathcal P. \end{align*}

Define the relative information function

\begin{align*} I_n^{\mathcal P\mid\mathcal G}:X&\to[0,\infty] \end{align*}

by

\begin{align*} I_n^{\mathcal P\mid\mathcal G}(x):=I_\mu(\mathcal P_{[0,n-1]}\mid\mathcal G)(x). \end{align*}

Then

\begin{align*} \frac{1}{n}I_n^{\mathcal P\mid\mathcal G}(x)\to h_\mu(T,\mathcal P\mid \mathcal G) \end{align*}

in $L^1$ and for $\mu$-a.e. $x$.