According to Theorem 1 of [1], the stastistic $(X_1,\cdots,X_n)$ is minimal sufficient for the statistical model $X_1,\cdots,X_n\sim N(\theta,1)$ iid and $\theta\in\mathbb{R}$. This is false, as you can see in the question: Function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e..
However, this paper was written by a reputable author and that theorem is also corroborated by Theorem 1 and 4 of [2] and [3], respectively.
Before reading those papers in order to find an error, I want to confirm that I'm applying those theorems correctly.
According to the Theorem 1 of [1] we have:
Let $n\in\mathbb{N}^\times$ and $\{P_\theta:\mathcal{A}_\theta \to \mathbb{R}\}_{\theta\in \Theta}$ be a family of perfect probability measures admitting a countable generated sufficient $\sigma$-algebra. If $T$ is a minimal sufficient statistic for that family, then $S$ given by $S(x_1,\cdots,x_n):=(T(x_1),\cdots,T(x_n))$ is a minimal sufficient statistic for the product family $\{\bigotimes_{i=1}^n P_\theta:\bigotimes_{i=1}^n \mathcal{A}_\theta \to \mathbb{R}\}_{\theta\in \Theta} $
For all $\theta\in\mathbb{R}$ let $P_\theta:\mathfrak{B}_\mathbb{R}\to \mathbb{R}$ be the probability measure induced by the density $f_\theta (x):=(2\pi)^{-1/2}e^{-\frac{(x-\theta)^2}{2}}$ (normal distribution).
Using the Neyman-Fisher Factorization Theorem we can prove that $T:\mathbb{R}\to \mathbb{R}$ given by $T(x):=x$ is a sufficient statistic w.r.t. $\{P_\theta:\mathfrak{B}_\mathbb{R}\to \mathbb{R}\}_{\theta\in\mathbb{R}}$. Since $\sigma(T)$ is countably generated (because $\mathbb{R}$ is the codomain of $T$), we can conclude that $\{ P_\theta:\mathfrak{B}_\mathbb{R}\to \mathbb{R}\}_{\theta\in\mathbb{R}}$ admits a countable generated sufficient $\sigma$-algebra.
According to 7.1.7 and 7.5.10 Theorems of "Measure Theory" written by V.I. Bogachev, the $P_\theta$ is a perfect probability measure for all $\theta\in\mathbb{R}$.
Therefore, applying the previous theorem we conclude that $S:\mathbb{R}^n\to\mathbb{R}^n$ given by $S(x_1,\cdots,x_n):=(T(x_1),\cdots,T(x_n))=(x_1,\cdots,x_n)$ is minimal sufficient statistic for $\{\bigotimes_{i=1}^n P_\theta:\mathfrak{B}_{\mathbb{R}^n}\to \mathbb{R}\}_{\theta\in\mathbb{R}}$.
However, using the Neyman-Fisher Factorization Theorem, we can conclude that $R:\mathbb{R}^n\to \mathbb{R}$ given by $R(x_1,\cdots,x_n):=\sum_{i=1}^nx_i$ is a sufficient statistic for $\{\bigotimes_{i=1}^n P_\theta:\mathfrak{B}_{\mathbb{R}^n}\to \mathbb{R}\}_{\theta\in\mathbb{R}}$.
With the previous results we arrive at a contraction, as you can see in the question I mentioned before.
My question is: Did I make a mistake earlier or were the conditions of that theorem correctly met? Does anyone have any idea where the mistake is in those papers?
