Conditional probability, statistic and sufficient statistic In statistical model $(\mathcal{X}, \{P_\theta\mid\theta\in\Theta\})$ statistic $T=T(\mathbf{X})$ (where $\mathbf{X}$ marks random sample) is said to be sufficient for $\theta$, when conditional distribution of random sample with respect to $T$ does not depend on $\theta$. In modern probability approach conditional probability is defined in terms of condinal expectations, i.e.
$$
P_\theta(A\mid\ T)=P_\theta(A\mid\sigma(T))\equiv \mathbb{E}_\theta[\mathbb{1}_A(\mathbf{X})\mid 
T],\quad\forall\theta\in\Theta$$where $A\in\mathcal{B}(\mathbb{R}^n)$. It means that $T$ is sufficient statistic if and only if conditional probability is a statistic (function of random sample, which does not depend on $\theta).$ If someone meet the definition like this somewhere? Maybe it's something wrong with it? I'm just curious. 
 A: If you're asking for a rigorous definition of sufficiency using modern probability theory/measure theory, then one of the following definitions (in decreasing levels of generality) might be what you're looking for.

Definition 1. (Sufficient sub-$\sigma$-algebra)
Let $(\mathcal{X}, \mathcal{B})$ be a measurable space (the sample space), and let $\mathcal{P}$ be a set of probability measures on $(\mathcal{X}, \mathcal{B})$ (the candidate distributions). Let $\mathcal{S}$ be a sub-$\sigma$-algebra of $\mathcal{B}$.
We say that $\mathcal{S}$ is sufficient for $\mathcal{P}$ if there exists a transition probability kernel $r : \mathcal{X} \times \mathcal{B} \to [0, 1]$ such that $r(\cdot, B)$ is a version of $P(B \mid \mathcal{S})$ for each $B \in \mathcal{B}$ and $P \in \mathcal{P}$.
That is, when conditioning on a sufficient $\sigma$-algebra $\mathcal{S}$, we no longer distinguish between the various candidate distributions $P \in \mathcal{P}$.
This is the meaning of the line "$r(\cdot, B)$ is a version of $P(B \mid \mathcal{S})$ for each $B \in \mathcal{B}$ and $P \in \mathcal{P}$."
Unwrapping this definition a bit, $\mathcal{S}$ is sufficient for $\mathcal{P}$ if and only if there exists a function $r : \mathcal{X} \times \mathcal{B} \to [0, 1]$ such that the following all hold.


*

*For each $B \in \mathcal{B}$, the function $x \mapsto r(x, B)$ from $\mathcal{X}$ into $[0, 1]$ is $\mathcal{S}$-measurable.

*For each $x \in \mathcal{X}$, the function $B \mapsto r(x, B)$ from $\mathcal{B}$ into $[0, 1]$ is a probability measure on $(\mathcal{X}, \mathcal{B})$.

*For all $B \in \mathcal{B}$, $S \in \mathcal{S}$, and $P \in \mathcal{P}$, we have
$$
P(B \cap S) = \int_S r(\cdot, B) \, dP.
$$

Definition 2. (Sufficient statistic)
Let $(\mathcal{X}, \mathcal{B})$ be a measurable space (the sample space), and let $\mathcal{P}$ be a set of probability measures on $(\mathcal{X}, \mathcal{B})$ (the candidate distributions). Let $(\mathcal{T}, \mathcal{C})$ be another measurable space.
A statistic $T : \mathcal{X} \to \mathcal{T}$ is sufficient for $\mathcal{P}$ if the generated $\sigma$-algebra
$$
\sigma(T) = \big\{\{T \in C\} : C \in \mathcal{C}\big\}
$$
is sufficient for $\mathcal{T}$ in the sense of Definition 1.
We can also unpack Definition 2.
A statistic $T$ is sufficient for $\mathcal{P}$ if and only if there exists a function $r : \mathcal{X} \times \mathcal{B} \to [0, 1]$ such that the following all hold.


*

*For each $B \in \mathcal{B}$, the function $x \mapsto r(x, B)$ from $\mathcal{X}$ into $[0, 1]$ is $\sigma(T)$-measurable.

*For each $x \in \mathcal{X}$, the function $B \mapsto r(x, B)$ from $\mathcal{B}$ into $[0, 1]$ is a probability measure on $(\mathcal{X}, \mathcal{B})$.

*For all $B \in \mathcal{B}$, $C \in \mathcal{C}$, and $P \in \mathcal{P}$, we have
$$
P(B \cap \{T \in C\}) = \int_{\{T \in C\}} r(\cdot, B) \, dP.
$$
In some cases (e.g., when $(\mathcal{X}, \mathcal{B})$ and $(\mathcal{T}, \mathcal{C})$ are standard Borel spaces, which includes essentially all spaces in practice), $T$ being sufficient for $\mathcal{P}$ is equivalent to the existence of a function $r^\prime : \mathcal{T} \times \mathcal{B} \to [0, 1]$ satisfying the following properties.


*

*For each $B \in \mathcal{B}$, the function $t \mapsto r^\prime(t, B)$ from $\mathcal{T}$ into $[0, 1]$ is $\mathcal{C}$-measurable.

*For each $t \in \mathcal{T}$, the function $B \mapsto r^\prime(t, B)$ from $\mathcal{B}$ into $[0, 1]$ is a probability measure on $(\mathcal{X}, \mathcal{B})$.

*For all $B \in \mathcal{B}$, $C \in \mathcal{C}$, and $P \in \mathcal{P}$, we have
$$
P(B \cap \{T \in C\}) = \int_C r^\prime(\cdot, B) \, dT_*P,
$$
where $T_*P$ is the distribution of $T$ under $P$: $(T_*P)(C) = P(\{T \in C\})$ for all $C \in \mathcal{C}$.
Heuristically, $r^\prime(t, B) = \mathbf{P}(X \in B \mid T = t)$, where $X$ is the "data" and $\mathbf{P}$ is some underlying, "ground truth" probability measure: it does not depend on the set $\mathcal{P}$ of candidate distributions.

Now consider the parametric case. A parametric statistical model consists of


*

*a measurable space $(\mathcal{X}, \mathcal{B})$ (the sample space),

*a measurable space $(\Theta, \tau)$ (the parameter space),

*and a transition probability kernel $\Theta \times \mathcal{B} \to [0, 1]$, denoted $(\theta, B) \mapsto P_\theta(B)$ for $\theta \in \Theta$ and $B \in \mathcal{B}$, which is the parametrization of the sampling distribution.


In this setup, we can define the obvious family of candidate distributions $\mathcal{P} = \{P_\theta : \theta \in \Theta\}$, and then the two definitions of sufficiency given above carry over unchanged.
For example, a statistic $T : \mathcal{X} \to \mathcal{T}$ is sufficient for $\mathcal{P}$ (or by slight abuse of notation, for $P$ or $\theta$ or $P_\theta$) if and only if there exists a function $r : \mathcal{X} \times \mathcal{B} \to [0, 1]$ such that the following hold.


*

*For each $B \in \mathcal{B}$, the function $x \mapsto r(x, B)$ from $\mathcal{X}$ into $[0, 1]$ is $\sigma(T)$-measurable.

*For each $x \in \mathcal{X}$, the function $B \mapsto r(x, B)$ from $\mathcal{B}$ into $[0, 1]$ is a probability measure on $(\mathcal{X}, \mathcal{B})$.

*For all $B \in \mathcal{B}$, $C \in \mathcal{C}$, and $\theta \in \Theta$, we have
$$
P_\theta(B \cap \{T \in C\}) = \int_{\{T \in C\}} r(\cdot, B) \, dP_\theta.
$$
In particular, $P_\theta(B \mid T) = r(\cdot, B)$ for all $\theta \in \Theta$, so when we condition on $T$, all $P_\theta$'s become the same.
