Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\sigma$-field $\cal{F}$. A sufficient statistic is defined to be an $\cal{F}$-measurable mapping $T:\cal{X}\rightarrow \mathbb{R}$ such that $P_{\theta}(X\mid T=t)$ does not depend on the $\theta\in\Theta$.

According to Neyman-Fisher factorization theorem, the sufficient and necessary condition that $T$ is a sufficient statistic for $\cal{P}$ is that $$p(X\mid \theta)=g(T,\theta)h(X),\forall\theta \in \Theta,X\in\cal X$$ where $p(X\mid \theta)=\frac{dP_{\theta}}{d\lambda}$ is the density/Radon-Nikodym derivative w.r.t. $\lambda$.

My questions:

(1) Since the conditional probability measure $P_{\theta}(X\mid T)$ is defined as a solution to the functional equation $\int_{B}P_{\theta}(A \mid \sigma(T))dP_{\theta}=P_{\theta}(A\cap B), \forall B\in \sigma(T)$, are there any characterization of the notion of sufficiency in terms of the $\sigma$-field $\sigma(T)$ generated by the $\cal{F}$-measurable mapping $T$ ,AND the $\sigma$-fields associated with $\cal{P}$?

(2) What is the implication of Neyman-Fisher Theorem in terms of the product measure $\lambda_\Theta\times\lambda$, assuming there is some measure $\lambda_\Theta$ given on the parameter space $\Theta$? It doe not seem like "independence" though.

I did not post this on stats.SE because I think it will receive better response here.