I am interested in dynamical gadgets which can be described by sampling along the orbits of points in some ergodic system $(\Omega,\mu,T)$. When $\mu$ is a probability measure, the theory of such objects is quite well understood, so I would like to study what happens when one allows the measure $\mu$ to be a properly $\sigma$ - finite measure (i.e. infinite and $\sigma$-finite). In this setting, a $\mu$-preserving transformation $T:\Omega \to \Omega$ is called ergodic if $T^{-1}E=E \Longrightarrow \mu(E) = 0 $ or $\mu (\Omega - E) = 0$.

In this setting, some facts from finite ergodic theory carry over - for example, one still has that any $T$-invariant measurable function $\Omega \to \mathbb{R}$ must be $\mu$-almost surely constant. However, some of the nicer results, such as the Birkhoff ergodic theorem fail to be true.

In the probability measure setting, we have the following very pleasant subadditive ergodic theorem, due to Kingman:

Theorem (Kingman): Suppose $(\Omega,\mu,T)$ is ergodic, and $f_n:\Omega \to \mathbb{R}$ are measurable, obey the subadditivity condition $f_{n+m}(x) \leq f_n(x)+f_m(T^n(x))$, and satisfy $\|f_n\|_{\infty} \leq C\cdot n$. Then the limit

$\displaystyle\lim_{n \to \infty}\frac{1}{n}\int_{\Omega} f_n(x) d \mu(x) $

exists and is equal to $\lim_{n \to \infty}\frac{1}{n} f_n(x) $ for $\mu$ almost every $x \in \Omega$.

My question is the following: does Kingman carry over to the infinite measure setting? If not, are there any weakened generalizations that one can obtain if one makes additional "niceness" assumptions about $\mu$ and\or $T$?