I figured it out by looking at the last few chapters in [Shiryayev] and some thoughts. The problem can be considered in following way with the aid of a formalized definition of conditional expectation.
Without loss of generality we assumed the sample random variable $X$ from the model $\cal{P}$ to be $L^1(P),\forall P\in \cal{P}$. This is easily verified to be true for any $P$ dominated by a $\sigma$-finite measure $\mu$ on the probability space $(\Omega,\mathcal{F},P)$. Also we discuss the situation where $X,T\geq 0,\forall\omega\in\Omega $. Here $$X:\Omega\rightarrow\mathbb{R},\omega\mapsto X(\omega)$$ and $$T:\Omega\rightarrow\mathbb{R},\omega\mapsto T(X(\omega))$$ are random variables. $X$ is the sample random variable; $T$ is the sufficient statistics/random variable.
Invoke Radon-Nikodym Theorem on the measure defined by $\nu(G):=\int_{G}XdP$ on subspace $(\Omega,\sigma(T),P)$ where $\sigma(T)\subset\cal{F}$ is the $\sigma$-field generated by sufficient statistic $T$, then the R-N derivative $\frac{d\nu}{d\mu}$ is the conditional expectation (random variable) $\boldsymbol{E}(X\mid T)(\omega)$. This can be directly checked: $$\int_{G}\boldsymbol{E}(X\mid T)(\omega)d\mu(\omega)$$ $$=\int_{G}\frac{d\nu}{d\mu}d\mu(\omega)=\int_{G}d\nu(\omega)=\nu(G)$$ for any $G\in \sigma(T)$. With this definition, we can directly assert that a conditional expectation is a simple random variable if $A_n$ is a partition of $\Omega$, i.e. $\cup_n A_n=\Omega,A_n\in \sigma(T)$ such that (since we want $1_{A_n}$ to be $\sigma(T)$-measurable functions.) $$\boldsymbol{E}(X\mid T)(\omega)=\frac{\nu(A_n)}{P(A_n)}1_{A_n}(\omega)$$
We know that if a sufficient statistics $T$ for model $\mathcal{P}=\{P\}$ , then from the expression above, the partition $\{A_n\}$ will satisfy that $\frac{\nu(A_n)}{P(A_n)}=\frac{\int_{A_n}XdP}{P(A_n)}=\frac{\int_{A_n}XdP}{\int_{A_n}1dP}$ is independent of the choice of $P\in\cal{P}$.
The level sets of $T$ form a partition as well as generating class(not necessarily atoms) of $\sigma(T)$. So a sufficient statistic will have its level sets partition the $\Omega$ in such a way that the conditional expectation is a constant random variable on each partition component independent of the choice of $P$. In other words, $T$ is a "piece-wise" constant set function, being constant on those sets $A_n\in\cal{F}$ such that $P\{A_n\}\equiv C_n\in\mathbb{R},\forall P\in\cal{P}$
The minimal sufficient statistics $U=\phi(T)$ for some measurable function $\phi$ because $\sigma (U)$ is the coarest partition possible satisfying above requirements. i.e. $\sigma(U)\subset\sigma(T)$ so $U$ must be $\sigma(T)$-measurable, and the relation $U=\phi(T)$ follows from the fact that any $\sigma(T)$-measurable random variable $U$ can be written as $\phi(T)$ for some Borel function $\phi$.