Notation i.i.d sample I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the estimation of a distribution from an i.i.d. sample.
Let $(\Omega,\mathcal{A},P)$ be a probability space and $(E,\mathcal{E})$, $(E_1,\mathcal{E}_1),\dotsc,(E_n,\mathcal{E}_n)$ be measurable spaces.
The random variable $X\colon \Omega\to E$ is distributed according to $P_X$. 
Let $X_i\colon \Omega\to E_i$ also be random variables and define a sample as
$Z=(X_1,\dotsc,X_n)\colon \Omega\to E_1 \times \cdots \times E_n$ as a random varible with measurable space $(\times_{i=1}^n E_i, \otimes_{i=1}^n \mathcal{E}_i)$.
If the samples are independent, then $P_Z = P_{X_1} \otimes \cdots \otimes P_{X_n}$.
How do I correctly say that I am interested in the estimation of $P_X$, but have only access to independent observations from this distribution? Is in this case 


*

*$P_Z = P_{X} \otimes \cdots \otimes P_{X}$

*$(\times_{i=1}^n E_i, \otimes_{i=1}^n \mathcal{E}_i) = (\times_{i=1}^n E, \otimes_{i=1}^n \mathcal{E})$?


Furthermore, let $Z=z=(x_1,\dotsc,x_n)$ be a realization of the sample.
Given the empirical distribution $$P_n(E) = n^{-1}\sum_{i=1}^{n} \mathbb{1}_E \circ X_i(\omega), \quad E\in\mathcal{E}, \omega \in \Omega$$
then
$$
P_Z\bigg( \Big\{\lim_{n\to\infty} P_n(E) = P_X(E) \Big\} \bigg) = 1, \quad E \in \mathcal{E} \quad \text{or}\\
P\bigg( \Big\{\lim_{n\to\infty} P_n(E)(\omega) = P_X(E) \Big\} \bigg) = 1, \quad E \in \mathcal{E}, \omega \in \Omega
$$
Is the notation above formally correct?
 A: I post here the corrected example.
Let $(\Omega,\mathcal{A},P)$ be a probability space and $(E,\mathcal{E})$, $(E_1,\mathcal{E}_1),\dotsc,(E_n,\mathcal{E}_n)$ be measurable spaces.
The random variable $X\colon \Omega\to E$ is distributed according to $P_X$. 
Let $X_i\colon \Omega\to E_i$ also be random variables and define a sample as
$Z=(X_1,\dotsc,X_n)\colon \Omega\to E_1 \times \cdots \times E_n$ as a random varible with measurable space $(\times_{i=1}^n E_i, \otimes_{i=1}^n \mathcal{E}_i)$.
If the samples are independent, then $P_Z = P_{X_1} \otimes \cdots \otimes P_{X_n}$.
In this case


*

*$P_Z = P_{X} \otimes \cdots \otimes P_{X}$

*$(\times_{i=1}^n E_i, \otimes_{i=1}^n \mathcal{E}_i) = (\times_{i=1}^n E, \otimes_{i=1}^n \mathcal{E})$


Furthermore, let $Z=z=(x_1,\dotsc,x_n)$ be a realization of the sample.
Given the empirical distribution $$P_n(E) = n^{-1}\sum_{i=1}^{n} \mathbb{1}_E \circ X_i(\omega), \quad E\in\mathcal{E}, \omega \in \Omega$$
then
$$
P\bigg( \Big\{\omega \in \Omega \mid \lim_{n\to\infty} P_n(E)(\omega) = P_X(E) \Big\} \bigg) = 1
$$
or alternatively,
$$
P\bigg( \lim_{n\to\infty} P_n(E) = P_X(E)  \bigg) = 1
$$
