Consider the probability space $(\mathcal{S}, \mathcal{A}, \mu)$. Suppose that the random variables $X_1,..., X_n$ are exchangeable and take values in the Borel space $(\mathcal{X}, \mathcal{B})$. We can prove that the empirical probability measure $P_n$ is a measurable function from $(\mathcal{X}^n, \mathcal{B}^n)$ to $(\mathcal{P}, \mathcal{C}_{\mathcal{P}})$, where $\mathcal{P}$ is the set of all probability measures on $(\mathcal{X}, \mathcal{B})$ (Schervish, Theory of Statistics, Sec. 1.4 Prob. 24.).
I have a problem in understanding where do we use exchangeability.
Indeed, suppose for simplicity $\mathcal{X}=\{0,1\}$.
$\forall x:=(x_1,..., x_n)\in \mathcal{X}^n$, we have
$$ \begin{cases} P_n(x)(\{1\}):=\frac{1}{n}\sum_{i=1}^n 1(x_i=1)\\ P_n(x)(\{0\}):=\frac{1}{n}\sum_{i=1}^n 1(x_i=0)\\ P_n(x)(\{1,0\}):=\frac{1}{n}\sum_{i=1}^n 1(x_i\in \{0,1\})\\ P_n(x)(\emptyset):=\frac{1}{n}\sum_{i=1}^n 1(x_i\in \emptyset)\\ \end{cases} $$ and hence $P_n(x):\mathcal{B}\rightarrow[0,1]$ is a probability measure on $(\mathcal{X}, \mathcal{B})$ assigning $P(x)(B)\in [0,1]$ to each $B\in \mathcal{B}$.
Hence, $P_n: \mathcal{X}^n\rightarrow \mathcal{P}$.
Or (equivalently?), suppose for simplicity $\mathcal{S}:=\{s_1,s_2,s_3\}$, $\mathcal{X}:=\{0,1\}$, $n=2$, $X_1(s_1)=1,X_1(s_2)=1 , X_1(s_1)=0$, and $X_2(s_1)=0,X_2(s_2)=1, X_2(s_1)=0$.
$\forall s:=(s_1,s_2)\in \mathcal{S}^2$, we have
$$ \begin{cases} P_n(s)(\{1\}):=\frac{1}{2}\sum_{i=1}^2 1(X_i(s_i)=1)\\ P_n(s)(\{0\}):=\frac{1}{2}\sum_{i=1}^2 1(X_i(s_i)=0)\\ P_n(s)(\{1,0\}):=\frac{1}{2}\sum_{i=1}^2 1(X_i(s_i)\in \{0,1\})\\ P_n(s)(\emptyset):=\frac{1}{2}\sum_{i=1}^2 1(X_i(s_i)\in \emptyset)\\ \end{cases} $$ and hence $P_n(s):\mathcal{B}\rightarrow[0,1]$ is a probability measure on $(\mathcal{X}, \mathcal{B})$ assigning $P(s)(B)\in [0,1]$ to each $B\in \mathcal{B}$.
Hence, $P_n: \mathcal{S}^n\rightarrow \mathcal{P}$.
Where do we use exchangeability?
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