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?

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?

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

$$ L(\theta; y) = g(T(y);\theta)h(y) x_{1,2} = \frac{-5 \pm \sqrt{5^2-12}}{6} $$

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?

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?