Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $$\mathcal I:=\left\{A\in\mathcal A:\tau^{-1}(A)=A\right\}$$ and $X_n:\Omega\to[-\infty,\infty)$ be $\mathcal E$-measurable for $n\in\mathbb N$ with $\operatorname E\left[X_1^+\right]<\infty$. Assume $$X_{m+n}\le X_m+X_n\circ\tau^m\;\;\;\text{for all }m,n\in\mathbb N\tag1.$$ It's then trivial to see that $\operatorname E\left[X_n^+\right]\le n\operatorname E\left[X_1^+\right]<\infty$ for all $n\in\mathbb N.$
I would like to show that $$\frac{\operatorname E\left[X_n\right]}n\xrightarrow{n\to\infty}\inf_{n\in\mathbb N}\frac{\operatorname E\left[X_n\right]}n\tag2$$ and $$\frac{\operatorname E\left[X_n\mid\mathcal I\right]}n\xrightarrow{n\to\infty}\inf_{n\in\mathbb N}\frac{\operatorname E\left[X_n\mid\mathcal I\right]}n\tag3\;\;\;\text{almost surely}.$$ Moreover, and this is the thing I'm struggling with most, $$\frac{X_n}n\xrightarrow{n\to\infty}\inf_{n\in\mathbb N}\frac{\operatorname E\left[X_n\mid\mathcal I\right]}n\;\;\;\text{almost surely}\tag4.$$
By Fekete's lemma, $$\frac{x_n}n\xrightarrow{n\to\infty}\inf_{n\in\mathbb N}\frac{x_n}n\in[-\infty,\infty)\tag5$$ for any subadditive $(x_n)_{n\in\mathbb N}\subseteq[-\infty,\infty)$. So, $(2)$ and $(3)$ are trivial from this result, since it's easy to see that $(\operatorname E[X_n])_{n\in\mathbb N}$ and $(\operatorname E[X_n\mid\mathcal I])_{n\in\mathbb N}$ are (pointwisely) subadditive.
Now, in light of the corresponding proof for the ordinary pointwise ergodic theorem, we might be able to utilize the subadditive version of the maximal ergodic theorem: $$\operatorname E\left[X_1;\max_{1\le i\le n}X_i\ge0\right]\ge0\;\;\;\text{for all }n\in\mathbb N.\tag6$$
I guess we are able to reduce the problem to the case where $Y_n\le0$ and $Y_n\in\mathcal L^1(\operatorname P)$ for all $n\in\mathbb N$. So, if you know a proof under this assumption, I'm interested in it as well.