Reference of the ergodic theorem in continuous time

Question

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\tau_t:\Omega\to\Omega$ for $t\ge0$ such that

1. $\tau_0=\operatorname{id}_\Omega$
2. $\tau_{s+t}=\tau_s\tau_t$ for all $s,t\ge0$
3. $\tau:\Omega\times[0,\infty)\to\Omega$ is $\left(\mathcal A\otimes\mathcal B\left([0,\infty)\right),\mathcal A\right)$-measurable
4. $\operatorname P\circ\:\tau_t^{-1}=\operatorname P$ for all $t\ge0$

Now, let $$\mathcal I:=\left\{A\in\mathcal A:\tau_t^{-1}(A)=A\text{ for all }t\ge0\right\}.$$

I'm searching for a reference (with proof) of the ergodic theorem of the following form: If $p\in[1,\infty)$ and $F\in\mathcal L^1(\operatorname P)$, then $$\frac1t\int_0^tF\circ\tau_t\:{\rm d}t\xrightarrow{t\to\infty}\operatorname E\left[F\mid\mathcal I\right]\tag1$$ almost surely and in $L^p(\operatorname P)$. I was only able to find either the discrete-time analogue or the special where $\tau$ is the time-shift on the canonical probability space of a stochastic process.

Answer 1

This answer comes late for @0xbadf00d but for others: a cts. time version of Birkhoff's Ergodic Theorem with proof can be found in Olav Kallenberg's Foundation of Modern Probability Volume 2 (3rd edition), p. 563.