Section 3 of Fortini et al. (2000) states that
Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not exist.[see e.g (4) in Section 48 of Halmos(1950)].
However, Halmos (1950) example is quite theoretical and shows it by contradiction. Does anyone know a simple canonical example, where $X$ and $P$ are explicitly defined?
To give more details, let $(X, \mathcal X)$ be a set and its $\sigma$-algebra, such that $(X^n, \mathcal X^n)$ is the $n$-fold Cartesian product of $X$ and its $\sigma$-algebra, where $n$ extends to $\infty$. We can assume that $\mathcal X^n =\mathcal{B}(X)^n$($\mathcal B$ for Borel). Let $P$ be a probability measure on $(X^\infty, \mathcal X^\infty)$ that makes a set of observations $(x_1, x_2, \dots, .. )\in X^\infty$ exchangeable.
Since one can define a predictive distribution of $x_n$ given $x_{(n-1)} = (x_1, \dots, x_{n-1})$ as a function $P_n$ with the following properties:
- (P1) $P_n(.;x_{(n-1)})$ is probability measure on $\mathcal X$.
- (P2) $P_n(A;.)$ is $\mathcal X^{n-1}$-measurable for each $A \in \mathcal A$.
- (P3) For every $A_1 \in \mathcal X^{n-1}$ and $A_2 \in \mathcal X$ we have $$ P(x_{(n)} \in A_1 \times A_2) = \int_{A_1} P_n(A_2, y)\mu(x_{(n-1})(dy)$$ where $\mu$ is the probability distribution of $x_{(n-1)}$.
To my understanding (correct me if I am wrong), a canonical example that shows for a given $(X^\infty, \mathcal X^\infty, P )$ no $P_n$ exists, might follow the lines of showing that one can find functions that satisfy conditions (P1) and (P2), but no function that satisfy (P3) as well.
We know that if $X$ is Polish than $P_n$ would always exists.
