A version of the disintegration theorem states:
Let $(X,\mathcal{A})$ be a standard Borel space. Let $p$ be a probability measure on $(X,\mathcal{A})$ $\mathcal{B}\subseteq\mathcal{A}$ be a sub-sigma-algebra. There exists a $\mathcal{B}$-measurably indexed family $y\mapsto p_y$ of probability measures on $(X,\mathcal{A})$ such that for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$, $$ \int_B p_y(A) \, p(dy) = p(A\cap B) , $$ i.e. such that $p_y(A)$ is a conditional expectation of the indicator function $1_A$ given $\mathcal{B}$.
Moreover, for any $B\in\mathcal{B}$, the function $y\mapsto p_y(B)$ is equal to the indicator $1_B(y)$ for $y$ on a subset $C_B$ of $\mathcal{B}$ of measure one for $p$.
I wonder:
- Can we strengthen the last sentence as follows? (Note that the quantifiers are different.)
Moreover, there is a subset $C$ of $\mathcal{B}$ of measure one for $p$ such that for all $y\in C$ and for all $B\in\mathcal{B}$, the function $y\mapsto p_y(B)$ is equal to the indicator $1_B(y)$.
This seems to be the case when $(X,\mathcal{B})$ is standard Borel as well, but we cannot assume that in general (in particular $\mathcal{B}$ might fail to separate points). What can we say in general?
- If the answer to the above is negative, can we still strengthen the last sentence as follows?
Moreover, there is a subset $C$ of $\mathcal{B}$ of measure one for $p$ such that for all $y\in C$ and for all $B\in\mathcal{B}$, the function $y\mapsto p_y(B)$ only assumes the values $0$ and $1$.
(Again, notice that we are asking $C$ not to depend on $B$.)