How to prove the Fisher-Neyman factorization theorem in the continuous case? [closed]

Question

Fisher -Neyman Factorization Theorem is:

A statistic $T(Y)$ is sufficient for $θ$ if and only if for all $θ\in Θ$ and all $y\in \Omega$ , there is

$$ L(\theta; y) = g(T(y);\theta)h(y) $$

where $g(.;.)$ depends on $T(y)$ and $\theta$, and $h(.)$ does not depend on $\theta$
My question is how to prove the Fisher-Neyman factorization theorem in the continuous case?

Answer 1

You find a proof f.i. in G.G. Roussas, A Course in Mathematical Statistics, 2. ed., Academic Press, 1997, Ch. 11, Th. 1 (p. 263).