Here we establish a general ergodic pathwise ergodic theorem and also consider the ergodic case in which the OP seems to be interested.
Concepts and definitions:
Suppose $(M,\mathscr{B},\mu)$ is a probability space ($M$ is a Polish space with a Borel $\sigma$-algebra for example). Let $\Omega=M^{\mathbb{Z}_+}$ equipped with the product $\sigma$-algebra $\mathscr{F}=\mathscr{B}^{\otimes\mathbb{Z}_+}$. For each $n\in\mathbb{Z}_+$ let $X_n:\Omega\rightarrow M$ be the projection $X_n(\omega)=\omega(n)$, define $\mathscr{F}_n:=\sigma(X_k:0\leq k\leq n)$. It is obvious that $(\mathscr{F}_n:n\in\mathbb{Z}_+)$ is a filtration and that the process $X:\omega\mapsto\omega$ us adapted to this filtration. There is (an applications if Ionescu-Tulcea's theorem for example) a unique probability measure $\mathbb{P}_\mu$ on $(\Omega,\mathscr{F})$ such that for any $A_0,\ldots, A_k\in\mathscr{B}$ and integers $0=n_0< n_1<\ldots <n_k$
$$\mathbb{P}_\mu[X_{n_j}\in A_j]=\int_{A_0}\int_{A_1}\ldots\int_{A_k}P^{n_k-n_{k-1}}(x_{k-1},dx_k)\ldots P^{n_1}(x_0,dx_1)\,\mu(dx_0)$$
where $P^0=I$ (identity) and $P^n=P P^{n-1}$ for $n\geq1$.
Under this probability, $X$ is time homogeneous Markov chain with initial probability $\mu$ with transitions kernel $P$. In particular, when $\mu=\delta_{x}$ for some $x\in M$ we use the notation $\mathbb{P}_x=\mathbb{P}_{\delta_x}$. It is easy to check that for any probability measure $\mu$ on $(M,\mathscr{B})$ and any bounded measurable function $F:(\Omega,\mathscr{F})\rightarrow(\mathbb{R},\mathscr{B}(\mathbb{R}))$
\begin{align}
\mathbb{E}_\mu[F]=\int_M\mathbb{E}_x[F]\,\mu(dx)\tag{0}\label{zero}
\end{align}
Let $\theta:\Omega\rightarrow\Omega$ define the shift operator $(\theta(\omega))(n)=\omega(n+1)$. Then, for any bounded measurable function $F:(\Omega,\mathscr{B})\rightarrow(\mathbb{R},\mathscr{B}(\mathbb{R})$
$$\mathbb{E}_\mu[F(X\circ\theta^m)|\mathscr{F}_m]=\mathbb{E}_{X_m}[F(X)]$$
It is easy to check that $\mu P=\mu$ iff $\theta$ is $\mathbb{P}_\mu$ invariant ($\mathbb{P}_\mu[\theta^{-1}(B)]=\mathbb{P}_\mu[B]$ for all $B\in\mathscr{F}$) or equivalently, $\mu P=\mu$ iff $X$ is stationary w.r.t $\mathbb{P}_\mu$. The following result is what the OP seems to be looking for:
Theorem PW: Suppose $\mu P=\mu$, $\mu(M)=1$. For any $f\in L_1(\mu)$ there is $B_f\in\mathscr{B}$ such that $\mu(B_f)=1$, and a function $f^*\in L_1(\mu)$ such that for all $x\in B_f$
$$\frac1n\sum^{n-1}_{k=1}f(X_k)\xrightarrow{n\rightarrow\infty}f^*(X_0)\qquad \text{$\mathbb{P}_x$-a.s}$$
Moreover, $\int f^* \,d\mu=\int f\,d\mu$. If $\mu$ is $P$-ergodic, then for all $x\in B_f$, $f^*=\int f\,d\mu$ $\mathbb{P}_x$-a.s
Ergodic theorems:
Recall that a $P$ invariant measure $\mu$ is $P$ ergodic if for for any absorbent set $A$ ($P\mathbb{1}_A\geq\mathbb{1}_A$), $\mu(A)\in\{0,1\}$.
In the particular case where $Pf=f\circ T$ for some $\mu$-invariant transformation $T$, $\mu$ is ergodic if $\mu(A)\in\{0,1\}$ for all $A\in\mathscr{B}$ with $T^{-1}(A)=A$.
It can be shown that if $\mu$ is $P$ ergodic, then for any $f\in L_1(\mu)$, $Pf=f$ $\mu$-a.s iff $f=\mu[f]:=\int f\,d\mu$ $\mu$-a.s.
We have state two ergodic theorems that act on different types of transformation. The first one is a direct consequence of the ergodic theorem of Dunford-Hopf-Schwartz and von Neumann for positive contractions:
Theorem DHS: Suppose $\mu P=\mu$, $\mu(M)=1$. For any $1\leq p<\infty$ and $f\in L_p(\mu)$ there is $Af\in L_p(\mu)$ such that
$$A_nf=\frac{1}{n}\sum^{n-1}_{k=0}P^kf\xrightarrow{n\rightarrow\infty}Af$$
$\mu$-a.s. and in $L_p(\mu)$. Furthermore, $P(Af)=Af=A(Pf)$ $\mu$-a.s., and $\mu[Af]=\mu[f]$. If $\mu$ is ergodic, then $Af=\mu[f]$ $\mu$-a.s.
- The $\mu$-a.s. convergence is known as individual ergodic theorem; convergence in $L_p$ is known as the mean ergodic theorem (von Neumann).
- The limit $Af$ can be expressed in probabilistic terms. Let $\mathcal{I}^P_\mu$ the collection of sets in $\mathscr{B}$ such that
$P\mathbb{1}_B=\mathbb{1}|_B$ $\mu$-a.s. It is possible to prove that $\mathcal{I}^p_\mu$ is a $\sigma$-algebra and that $Af=\mu[f|\mathcal{I}^P_\mu]$ $\mu$-a.s.(conditional expectation of $f$ given $\mathcal{I}^p_\mu$ under $\mu$).
On the other hand, we have the well known ergodic theorem of Birkoff and von Neumann
Theorem B: Suppose $(S,\mathscr{S},m)$ is a probability space and $T:(S,\mathscr{S})\rightarrow(S,\mathscr{S})$ is $\mu$ invariant ($m(T^{-1}(B))=m(B)$ for all $B\in\mathscr{S})$. For $1\leq p<\infty$ and $f\in L_p(m)$ there is $f^*\in L_p$ such that
$$\frac{1}{n}\sum^{n-1}_{k=0}f\circ T^k\xrightarrow{n\rightarrow\infty} f^*$$
$m$-almsot surely and in $L_p(m)$. In particular, $$m[f^*]=m[f]$$ If $m$ is $T$ ergodic, $f^*=m[f]$ $m$-a.s.
As with Theorem DHS, $f^*$ can be expressed as a conditional expectation. Let $I_m=\{A\in\mathscr{S}: \mu(T^{-1}(A)\triangle A)=0$. this is a $\sigma$-alsgebra and $f^*=m(f|\mathcal{I}_\mu)$ $\mu$-a.s.
The Birkoff ergodic theorem can be obtained from the stronger DHS Theorem by cosidering the transition function $P_Tf:=f\circ T$ for all bonded measurable functions $f$ in $(S,\mathscr{S})$. This however might be an overkill. The technique of maximal inequalities can be used to prove both theorems (the individual parts); the $L_2$ result of von Neumann's theorem can be exploited to prove the mean ergodic results in borsht theorems.
We will only use explicitly Birkoff's ergodic theorem in the rest of this posting.
Proof of path wise ergodic theorem:
Applying Birkoff's ergodic theorem with $(S,\mathscr{S},m,T)=(\Omega,\mathscr{F},\mathbb{P}_\mu,\theta)$ we have that for any $G\in L_1(\mathbb{P}_\mu)$ there is $G^*\in L_1(\mathbb{P}_\mu)$ with $G^*\circ\theta=G^*$, $\mathbb{E}_\mu[G^*]=\mathbb{E}_\mu[G]$ and
\begin{align}\frac1n\sum^{n-1}_{k=0}G\circ\theta^k\xrightarrow{n\rightarrow\infty}G^*\tag{1}\label{one}\end{align}
$\mathbb{P}_\mu$-a.s. and in $L_1(\mathbb{P}_\mu)$.
Lemma: Suppose that $F\in\mathscr{F}_\infty$ is a bounded $\theta$--invariant function, that is $F=F\circ\theta$. If $\mathbb{P}_\mu$ is $\theta$ invariant, then $F=\mathbf{E}_\mu[F|\mathscr{F}_0]$ $\mathbb{P}_\mu$--a.s.
Proof of Lemma: Suppose
$F\circ\theta=F$ and define $h(x)=\mathbb{E}_x[F]$. Then
$$h(X_k)=\mathbb{E}_{X_k}[F]=\mathbb{E}_\mu[F\circ\theta^k|\mathscr{F}_k]=\mathbb{E}_\mu[F|\mathscr{F}_k]$$
It follows that $(h(X_k):k\in\mathbb{Z}_+)$ is a uniform integrable martingale with respect to the filtration $(\mathscr{F}_j:k\in\mathbb{Z}_+)$. An application of the martingale convergence theorem implies that
$$h(X_k)\xrightarrow{k\rightarrow\infty} \mathbb{E}[F|\mathscr{F}_\infty]=F$$
$\mathbb{P}_\mu$-a.s. and in $L_1(\mathbb{P}_\mu)$. Using the stationarity of $\mathbb{P}_\mu$ again yields
\begin{align}
\|\mathbb{E}_{\mu}[F|\mathscr{F}_0]-F\|_{L_1(\mathbb{P}_\mu)}=
\|(h(X_0) -F)\circ\theta^k\|_{L_1(\mathbb{P}_\mu)}=
\|h(X_k)-F\|_{L_1(\mathbb{P}_\mu)}\xrightarrow{k\rightarrow\infty}0
\end{align}
Therefore $\mathbb{E}[F|\mathcal{F}_0]=F$ $\mathbb{P}_\mu$-a.s. $ \blacksquare $
The Lemma above shows that $F^*=\mathbb{E}[F^*|\mathbf{F}_0]$ $\mathbb{P}_\mu$-a.s. and thus, there is a function $f^*\in L_1(\mu)$ such that $F^*(X)=f^*(X_0)$ $\mathbb{P}_\mu$-a.s.
Now, for $f\in L_1(\mu)$ and define consider $F_f(\omega):=f(\omega_0)$. Then, for some $f^*\in L_1(\mu)$,
\begin{align}\frac1n\sum^{n-1}_{k=0}f(X_k)\xrightarrow{n\rightarrow\infty}f^*(X_0)\tag{2}\label{two}\end{align}
$\mathbb{P}_\mu$-a.s. and in $L_1(\mathbb{P}_\mu)$ . Hence
$$\int\mathbb{P}_x\Big[\{\frac1n\sum^{n-1}_{k=0}f(X_k)\xrightarrow{n\rightarrow\infty}f^*(X_0)\big\}\Big]\,\mu(dx)=1$$
It follows that there is $B_f\in\mathscr{B}$ such that
$\mu(B_f)=1$, and for $x\in B_f$
\begin{align}
\frac1n\sum^{n-1}_{k=0}f(X_k)\xrightarrow{n\rightarrow\infty}f^*(X_0) \qquad\text{$\mathbb{P}_x$-a.s.} \tag{3}\label{three}
\end{align}
Ergodic case
When $\mu$ is $P$ ergodic, the limit function $f^*$ in the pathwise ergodic theorem described above is constant ($\mathbb{P}_x$-a.s. for all $x\in B_f$). This seems to be the result that the OP is mostly interested. This follows directly from the following result:
Theorem E: $\mu$ is $P$ ergodic iff $\theta$ is $\mathbb{P}_\mu$ ergodic.
Proof of Theorem E:
Necessity: Suppose $\mu$ is $P$ ergodic. For any bounded $\mathscr{F}$--measurable
function $F$ define $h: x\mapsto \mathbb{E}_x[F]$. The Markov property implies that
\begin{align}
\mathbb{E}_x[F\circ\theta^k] =\mathbb{E}_x[\mathbb{E}_{X_k}[F]] =\mathbb{E}_x[h(X_k)]=P^kh(x)
\end{align}
If $F=F\circ\theta$ then $Ph=h$ and so, $h=\int f\,d\mu=\mathbb{E}_\mu[F]$ $\mu$-a.s.
We will show that $F=\mathbb{E}_\mu[F]$ $\mathbb{P}_\mu$--a.s. Indeed,
let $A\in\mathcal{F}_n$. Using the Markov property once more yields
\begin{align}
\mathbb{E}_\mu[F;A]&=\mathbb{E}_\mu[F\circ\theta^n;A]=
\mathbb{E}_\mu[\mathbb{1}_A\mathbb{E}_\mu[F\circ\theta^n|\mathscr{F}_n]]\\
&=
\mathbb{E}_\mu[\mathbb{E}_{X_n}[F];A]=\mathbb{E}_\mu[h(X_n);A]=
\mathbb{E}_\mu[F]\mathbb{P}_\mu[A];
\end{align}
where the last equality follows from
$h=\mathbb{E}_\mu[F]$,,$\mu$--a.s. and the fact that the distribution of $X_n$ is $\mu$. Since $\mathscr{F}=\sigma(\cup_n\mathscr{F}_n)$ and $\cup_n\mathscr{F}_n$ is an algebra, it follows by monotone class arguments that $F=\mathbb{E}_\mu[F]$\ $\mathbb{P}_\mu$--a.s.
Sufficiency: Suppose that $\mathbb{P}_\mu$ is $\theta$--ergodic, and let $F$ be a bounded $\mathcal{F}$--measurable function. By Birkoff's ergodic theorem
\begin{align}
\frac1n\sum^{n-1}_{k=0}F\circ\theta^k \longrightarrow \mathbb{E}_\mu[F]\quad
\text{$\mathbb{P}_\mu$--a.s. and in $L_1(\mathbb{P}_\mu)$}
\end{align}
Consequently, there is $B\in\mathscr{B}$ with $\mu(B)=1$ such that
for all $x\in B$,
$\frac1n\sum^{n-1}_{k=0}F\circ\theta^k \longrightarrow \mathbb{E}_\mu[F]$
, $\mathbb{P}_x$--a.s. Thus, by dominated convergence,
\begin{align}
\frac1n\sum^{n-1}_{k=0}\mathbb{E}_x[F\circ\theta^k]
\longrightarrow \mathbb{E}_\mu[F],
\qquad \text{$\mu$--a.s.}\tag{4}\label{four}
\end{align}
Let $A$ be a $P$--absorbent set and
$F(\omega)=\mathbb{1}_A(\omega_0)$. Then, for any
$n\in\mathbb{Z}_+$, $\mathbb{E}_x[F\circ\theta^n]=\mathbb{E}_x[\mathbb{1}_A(X_n)]=P^n\mathbb{1}_A(x)\geq\mathbb{1}_A(x)$;
hence, $P^n\mathbb{1}_A=\mathbb{1}_A$ ,
$\mu$--a.s. and from \eqref{four},
$\mu[A]=\mathbb{1}_A$, $\mu$--a.s. Therefore, $\mu[A]\in\{0,1\}$.
$ \blacksquare $
Best Answer
$d$-dimensional, continuous ergodic theorem
If $g$ is stationary and integrable, then a continuous version of the ergodic theorem implies that \begin{equation*} \lim_{R \to \infty} \frac{1}{|B_{R}|} \int_{B_{R}} g(x,\omega) \, dx = \tilde{g}(\omega), \end{equation*} where $\tilde{g}$ is a $\{\tau_{x}\}_{x \in \mathbb{R}^{d}}$-invariant random variable (i.e. $\tilde{g}(\tau_{x}\omega) = \tilde{g}(\omega)$). When the group action is ergodic, $\tilde{g} = \mathbb{E}(g)$. Note that none of this requires the sub-additivity property you postulated concerning $g$.
On the contrary, the way to see the above is to define a function $A(\cdot,\omega)$ on measurable sets by $A(E,\omega) = \int_{E} g(x,\omega) \, dx$. This has the following additivity property: \begin{equation*} A(E,\omega) = \sum_{i = 1}^{N} A(E_{i},\omega) \quad \text{if} \, \, E = \bigcup_{i = 1}^{N} E_{i}, \, \, E_{i} \cap E_{j} = \phi. \end{equation*} Further, it is stationary and integrable in the sense that \begin{equation*} A(E + x, \omega) = A(E,\tau_{x}\omega), \quad \mathbb{E}(|A(B_{R},\omega)|) \leq C |B_{R}|. \end{equation*} Using these three properties, a continuous version of the ergodic theorem implies that \begin{equation*} \lim_{R \to \infty} \frac{1}{|B_{R}|} A(B_{R},\omega) = \tilde{g}(\omega). \end{equation*}
$d$-dimensional, continuous sub-additive ergodic theorem
As in the case of space averages, the $d$-dimensional version of sub-additivity I will mention uses sub-additivity over sets instead of points.
Let's go back a step and simply consider the $d$-dimensional sub-additive lemma. Here is a typical example: pick a constant $F$ and define the electrostatistic energy $E_{F}(U)$ in a domain $U \subseteq \mathbb{R}^{d}$ by \begin{equation*} E_{F}(U) = \inf \left\{ \int_{U} \left(\frac{1}{2} \|Du(x)\|^{2} + F u(x) \right) \, dx \, \mid \, u \in C^{\infty}_{c}(U) \right\}. \end{equation*} If $U = \cup_{i = 1}^{N} U_{i}$ and $\{U_{1},\dots,U_{N}\}$ is pairwise disjoint, then \begin{equation*} E_{F}(U) \leq \sum_{i = 1}^{N} E_{F}(U_{i}). \end{equation*} Working directly with cubes and mimicking the proof of Fekete's sub-additive lemma, one can show that there is a constant $\epsilon_{F} \geq 0$ such that \begin{equation*} \epsilon_{F} = \lim_{R \to \infty} \frac{1}{|Q_{R}|} E_{F}(Q_{R}) = \inf \left\{ \frac{E_{F}(Q_{R})}{|Q_{R}|} \, \mid \, R > 0 \right\}. \end{equation*} (Here $Q_{R}$ is the ball centered at $0$ with radius $R$ with respect to the $\ell^{\infty}$-norm, i.e. a cube centered at zero with sidelength $2R$. Actually, the Euclidean norm would give the same result, which is another good exercise.)
[Actually, one can show that $E_{F}(Q_{R}) = \epsilon_{F} R^{d}$, but that's not the point!]
The same thing is true in a random setting. Let $f : \mathbb{R}^{d} \times \Omega \to [0,\infty)$ be a stationary random field satisfying $\mathbb{E}\left(\int_{B(0,1)} |f(x)| \, dx\right) < \infty$. Imagine $f$ captures the microscopic properties of some material, which, in this case, vary from one place to another. This time, let's define the electrostatic energy $E(U,\omega)$ by \begin{equation*} E(U,\omega) = \inf \left\{ \int_{U} \left(\frac{1}{2} \|Du(x)\|^{2} - f(x) u(x) \right) \, dx \, \mid \, u \in C^{\infty}_{c}(U) \right\}. \end{equation*}
Notice that $E(,\omega)$ has the following sub-additivity property: if $U = \sum_{i = 1}^{N} U_{i}$, then \begin{equation*} E(U,\omega) \leq \sum_{i = 1}^{N} E(U_{i},\omega). \end{equation*} Additionally, it is stationary in the sense that \begin{equation*} E(U + x, \omega) = E(U,\tau_{x} \omega). \end{equation*} One can show that there is a $\tau$-invariant random variable $\tilde{\varepsilon}$ such that \begin{equation*} \tilde{\varepsilon} = \lim_{R \to \infty} R^{-d} E(Q(0,R),\omega), \end{equation*} where $Q(0,R)$ is the ball centered at $0$ with radius $R$ with respect to the $L^{\infty}$-norm. (One could replace $Q(0,R)$ by $B(0,R)$ with respect to any norm, in fact.) If you want, $\tilde{\varepsilon}$ is the macroscopic electrostatic energy density of the material.
When everything is ergodic, the density is constant and is given by \begin{equation*} \tilde{\epsilon} = \inf \left\{ \frac{\mathbb{E}(E(Q_{R}))}{|Q_{R}|} \, \mid \, R > 0\right\}. \end{equation*}
Reference:
The standard reference for this is Krengel's book Ergodic Theorems.