Let $(X, \mathcal X, \mathbb P)$ be a probability space, let $\mathcal G$ be a $\sigma$-subfield, and let $\mathbb P^{\mathcal G}$ be a regular conditional probability for $\mathbb P$ given $\mathcal G$. That is, for all $x \in X$, $\mathbb P_x^{\mathcal E}$ is a probability measure, $x \mapsto \mathbb P_x(A)$ is $\mathcal G$ measurable for all $A \in \mathcal X$, and $$\mathbb P(A \cap G) = \int_G \mathbb P^{\mathcal G}_x(A)\mathbb P(dx)$$ holds for all $A \in \mathcal X$ and $G \in \mathcal G$.
Is it true that $P_x^{\mathcal G} \ll \mathbb P$ for almost every $x$?
I can confirm that this is true if $\mathcal X$ is separable, but I'm not sure it holds in general.
Let $X=[0,1]^2$, $\mathcal X$ is its Borel algebra and $\mathcal G=\{[0,1]\times B : [0,1]\times B \in \mathcal X\}$. If $\mathbb P(A) = \lambda(\{a:(a,a)\in A\})$ for any $A\in \mathcal X$ (with $\lambda$ the Lebesgue measure on $[0,1]$) then for $(x,y)\in[0,1]^2$ we have $\mathbb P_{(x,y)}^{\mathcal G}(A)=1_{A}\{(y,y)\}$ and $\mathbb P(\{(0,0)\})=0\neq 1=\mathbb P_{(0,0)}^{\mathcal G}(\{(0,0)\})$.