TITLE: Fixed objects of the M endofunctor on category Meas
QUESTION [8 upvotes]: Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces. 
As Gerald Edgar & Michael Greinecker pointed out in this thread, there is the natural endofunctor $M : \operatorname{Meas} \to \operatorname{Meas}$ which sends a space $X$ to the collection $M(X)$ of (extended-real-valued) measures on $X$. This collection $M(X)$ is a measurable space, equipped with the minimal $\sigma$-algebra so that the evaluation functions $\mu \mapsto \mu(A)$ are measurable. A morphism $f : X \to Y$ is mapped to the push-forwarding map $f_* : \mu \mapsto \mu \circ f^{-1}$.
We may naturally iterate this endofunctor. Thinking of a measure on $X$ as a statistical ensemble, the space $M^2(X) = M(M(X))$ consists of ensembles-of-ensembles. Such hierarchical spaces are important in probability theory and dynamical systems. We may go further, defining $M^3(X) = M(M(M(X)))$ and so forth.
This simply generates a dynamical system on the category of measurable spaces, where the initial condition $X \in \operatorname{Meas}^{\operatorname{ob}}$ gets mapped to its successors $M(X)$, $M^2(X)$, $M^3(X)$, etc. Understanding these categorical dynamics is a hard problem, to say the least. Understanding the fixed ``points'', on the other hand, might actually be tractable. Hence the question:

What are the fixed objects of the endofunctor $M :\operatorname{Meas} \to \operatorname{Meas}$ ?

REPLY [2 votes]: Here is a candidate class of examples. I have made this community wiki so please feel free to edit it if you can answer it. Or, copy the text and make a new answer so we can give you reputation points.

Let $X_0 := X$ be any non-empty measurable space, and for each $n \in \mathbb N$, define the product $X_{n+1} := M(X_n) \times X_n$. This is a measurable space when equipped with the tensor product of $\sigma$-algebras. Define $X_{\infty}$ to be the projective limit of the sequence $X_0 \leftarrow X_1 \leftarrow X_2 \leftarrow \cdots$, where the arrows denote the projections onto the second components of the products. 
The existence of the limit gives a canonical section $X_{\infty} \to M(X_{\infty}) \times X_{\infty}$. By iterating this map with the projection onto the first component, we have defined a natural measurable function $\varphi : X_{\infty} \to M(X_{\infty})$.
Consequently, a point $x$ in $X_{\infty}$ contains the data of a measure $\varphi(x)$ on the space. It may be the case that this is all the data possessed by the point, in which case $X_{\infty} \cong M(X_{\infty})$. Is this the case? That is, 

is the function $\varphi : X_{\infty} \to M(X_{\infty})$ one-to-one and onto?

Note that the requirement that $X_0$ be non-empty is necessary. If $X_0 = \varnothing$, then $M(X_0) = \{0\}$ consists of the zero measure, but $X_1 = \{0\} \times \varnothing = \varnothing$. Consequently $X_{\infty} = \varnothing$ and $M(X_{\infty}) = \{0\} \ne X_{\infty}$.

This construction is based on the concept of an epistemic type space, which encodes the belief hierarchies of players in epistemic game theory.