Consider the Lorenz system with parameters $\sigma = 10$, $\rho = 28$, and $\beta = \frac{8}{3}$:
$$\dot{x} = 10(x - y)$$ $$\dot{y} = x(28 - z) - y$$ $$\dot{z} = xy - \frac{8}{3}z$$
I'm interested in calculating various "average" values related to this system. In order to find them, I'm interested to know if there's way of calculating the "physical" probability measure $\mathrm{d}\nu$ associated to the attractor. As pointed out in the comments, such a measure is known to exist. It is defined as a measure on $\mathbb{R}^3$ such that, for continuous* functions $h: \mathbb{R}^3 \to \mathbb{R}$, almost all solutions of the Lorenz system $u(t): \mathbb{R} \to \mathbb{R}^3$ near the attractor $A$ satisfy $$ \lim_{T \to \infty} \frac{1}{T} \int_0^T h(u(t)) \,\mathrm{d}t = \int_A h \,\mathrm{d}\nu $$
If I understand correctly, the measure $\mathrm{d}\nu$ can be understood via a measurable function $p: A \to \mathbb{R}$ such that for nice open subsets $U \subset \mathbb{R}^3$, with characteristic function $\chi_U$, $$ \lim_{T \to \infty} \frac{1}{T} \int_0^T \chi_U(u(t)) \, \mathrm{d}t = \nu(U \cap A) = \int_{U \cap A} p(x,y,z) \, \mathrm{d}V, $$ where $\mathrm{d}V$ is the Lebesgue measure on $A$.
My primary questions are: Is there a continuous representative of $p$? Can $p$ be calculated exactly, or only estimated? If it can be calculated, how is this done?
[My original motivation for this question was: I'm aware that for a finite discrete Markov system, if we construct a matrix $[p_{ij}]$ where $p_{ij}$ is the probability that the system transitions to state $i$ given that it is currently in state $j$, then in the long run the probability that at any given time the system is in state $i$ is the $i$th entry of a $1$-eigenvector of $[p_{ij}]$ (with some dependence on initial conditions determining which eigenvector, if there is more than one). I'm interested to know if there is a continuous analog of this vector, and how to calculate it if so.]
*The characteristic functions $\chi_U$ are not continuous, which may be an issue with the definition of $p$. Hopefully it's clear what properties I'm asking $p$ to satisfy even if there should be more restrictions on $U$ than just being an open set (e.g. $\overline{U \cap A} = \overline{U} \cap A$) or if there are other issues with using measurable, rather than continuous, $h$.
No such p can exist! The Lorenz system contracts phase space so its physical measure must be supported on a set with Lebesgue measure zero and therefore it cannot have a density function.
You can approximate the measure numerically though. The method you describe of using a continuous limit of a finite Markov chain is called Ulam's method.