Nature of the space of ontic states

I am studying the theoretical background of the famous Pusey-Barrett-Rudolph (PBR) theorem, specifically the difference between $\psi-$epistemic and $\psi$-ontic ontological models of Quantum Mechanics. In doing this I am following mainly the article by Harrigan and Spekkens: “Einstein, incompleteness and the epistemic view of quantum states". In this article, Harrigan and Spekkens assume the space of ontic states $\Lambda$ to be a measure space, i.e. with a defined measure $\lambda$, and they claim that an ontological model for quantum theory posits a function that assigns to every preparation procedure $P$ (or, equivalently, every quantum state, if we make the assumption that every quantum state corresponds exactly to a preparation procedure) a probability density function
\begin{equation*}
p(\cdot|P):\Lambda\to \mathbb{R}
\end{equation*}
(this is what they call “epistemic state", since it represents the observer's knowledge of the ontic state of the physical system)
Using this definition, they define an ontological model to be $\psi$-ontic if for every choice of quantum states $\psi$ and $\phi$, associated with the preparations $P_{\psi}$ and $P_{\phi}$ respectively, the supports of the epistemic states associated with these preparation procedures are disjoint (or, more in general, have null measure in $\Lambda$), that is:
\begin{equation*}
p(\lambda|P_{\psi})\cdot p(\lambda|P_{\phi})=0 \text{ }\forall \lambda\in \Lambda
\end{equation*}
However, I have also read another definition of ontological model (maybe more rigorous than that of Harrigan and Spekkens), which is the one given by Matthew Saul Leifer in his article “Is the quantum state real? An extended review of $\psi-$ontology theorems". In his definition of “ontological model", he claims that the space of ontic states $\Lambda$ is just a measurable space (i.e. it has not a define measure) and there exists a function $\Delta$ which assigns to every quantum state a probability measure $\mu$ on $\Lambda$.

On the one hand, I prefer Harrigan and Spekkens definition because it is more intuitive and practical to handle with the definitions of $\psi$-onticity and $\psi$-epistemicity of ontological models. On the other hand, I fear that assigning previously a measure on $\Lambda$ can lead to a loss of generality, so Leifer's definition seems to be more rigorous, but then the definitions of $\psi$-onticity and $\psi$-epistemicity become much more complicated and difficult to apply practically.

What is your opinion? What is the correct definition of ontological model for Quantum Mechanics?