What does the second index mean in probability distribution in finite events space definition In one of the books I have found such definition:
Let $\Omega$ be countable space 
$$\Omega=\{ \omega_1, \omega_2,... \}$$
For single element events function P is defined as follows:
$$ P(\{ \omega_i \})=p_i $$
where $p_i \geq 0 $ and  $\sum_i p_i=1$
Which is quite obvious. But I cannot fully understand the second part:
From the statements above, and probability axioms, results that if $A \subset \Omega$ and
  $$ A=\{\omega_{i_{1}},\omega_{i_{2}},...\} $$
then
$$P(A)=P(\{\omega_{i_{1}},\omega_{i_{2}},...\})=P(\{\omega_{i_{1}}\} \cup\{\omega_{i_{2}}\}\cup...)=$$
$$P(\{\omega_{i_{1}}\}) + P(\{\omega_{i_{2}}\})+...)=p_{i_1}+p_{i_2}+...$$
Although I understand the meaning of the definition, I don't quite get the idea of this second subscript. An "engineer friendly" (not mathematician) explanation would be great.
 A: If you are given a subset $A \subseteq \Omega$, you have naturally only some of the $\omega_i$ in $A$, some will not belong to $A$, for example $A = \{\omega_1, \omega_3, \omega_5, \ldots\}$. To denote a general such $A$ we have to possibilities:


*

*We can write $A = \{\omega_i \mid i \in I\}$ where $I \subseteq \mathbb N$ is the set of the indices of the $\omega_i$ belonging to $A$, in our above example $I =\{1,3,5, \ldots\}$.

*We can, as the author does, enumerate the set $I$ from 1., that is we write $I = \{i_1, i_2, i_3, \ldots\}$, we have in our example $i_1  = 1$, $i_2 = 3$, $i_3 = 5$, $\ldots$ Then $A$ becomes $A = \{\omega_{i_1}, \omega_{i_2}, \omega_{i_3}, \ldots \}$.
So: The $i_j$ are an enumeration of the indices of $A$'s elements, and $A = \{\omega_{i_j}\mid j \ge 1\}$ is the corresponding enumeration of $A$.
A: Assume one is interested in the set $A$ of the odd-numbered elements $\omega_i$ in $\Omega$, that is, $A=\{\omega_{2i-1}\mid i\geqslant1\}$. Then a representation of $A$ is $A=\{\omega_{i_j}\mid j\geqslant1\}$ with $i_j=2j-1$ for every $j\geqslant1$.
Note that this is not the only representation of $A$, for example $A=\{\omega_{i_j}\mid j\geqslant1\}$ with $i_{2j-1}=4j-1$ and $i_{2j}=4j-3$ for every $j\geqslant1$ (in words, enumerate 1,3,5,7,9,11,13,... as 3,1,7,5,11,9,13,...).
A: The subscripts $i_1,i_2,\ldots$ in $A=\{\omega_{i_1},\omega_{i_2},\ldots\}$ is just notation that indicates that $A$ contains some elements of $\Omega=\{\omega_1,\omega_2,\ldots\}$ but not necessarily all of them. 
For instance, we could have $i_n=2 n$ for $n=1,2,\ldots$, for which $i_1=2$, $i_2=4$, $i_3=6$ and so on. Then $A$ would consist of all $\omega$'s with even index, i.e. $$A=\{\omega_2,\omega_4,\omega_6,\ldots\}.$$
So the statement that 
$$
P(A)=p_{i_1}+p_{i_2}+\cdots
$$
just says that whenever you have a subset $A\subseteq \Omega$ containing the elements $\omega_{i_1},\omega_{i_2},\ldots$, then the probability of $A$ is just the sum of the $p_i$s corresponding to the elements of $A$.
