In my lecture notes, I am given the definitions for:
-the independence of two discrete random variables
-the independence of a set of discrete random variables
-the pairwise independence of a set of discrete random variables
When learning about probability generating functions, we have the theorem:
Let $X_1, X_2, \ldots$, be i.i.d. non-negative integer-valued discrete random variable and let $N$ be another non-negative integer-valued discrete random variable, independent from $\{X_1, X_2,\ldots\}$ then the probability generating function of the random sum
$$X_1 + X_2 + X_3 + \cdots + X_N$$
is $G_N(G_X(s))$.
My question is, what does it mean for N to be 'independent from $\{X_1, X_2,\ldots\}$'? Does it simply mean the set $\{N, X_1, X_2,\ldots\}$ is independent?
This is an introductory course in probability, please don't give a measure theoretic explanation.
$N$ being independent from $\{X_1,X_2,\ldots\}$ does not mean that $\{N,X_1,\ldots,\}$ is independent. Use your intuition about what "independent" should mean. Does $N$ being independent from some things mean those things are independent of eachother? No that doesn't sound quite right. If $N$ is independent of $X_1$ and $X_2$, it definitely means $N$ is independent of $X_1$ and $N$ is independent of $X_2$. But sometimes this is not enough as the information of $X_1$ and $X_2$ together may somehow give more information than expected. For instance, if I flip a coin a random number $N$ times, get $H$ heads and $T$ tails, knowing $H$ doesn't tell me much, knowing $T$ doesn't tell me much, but knowing $H$ and $T$ also tells me $N$. To account for this kind of scenario we must carefully choose our definition of independence to make sure that $N$ is independent from joint distributions of the $X_i$ too.
The definition of $N$ being independent from $\{X_1,X_2,\ldots\}$ is any one of the following equivalent conditions:
1) for any $n \in \mathbb{N}$ set of distinct indices $i_1,\ldots,i_n$ and any $t_0, t_1,\ldots, t_n \in \mathbb{R}$ we have $$P(N \leq t_0, X_{i_1} \leq t_1, \ldots, X_{i_n} \leq t_n) = P(N\leq t_0) P(X_{i_1} \leq t_1, \ldots, X_{i_n} \leq t_n).$$
Note: we do not split $P(X_{i_1} \leq t_1, \ldots, X_{i_n} \leq t_n)$ into $P(X_{i_1} \leq t_1)\cdots P(X_{i_n} \leq t_n)$ unless we are also requiring all the $X_i$ to be independent of eachother.
2) for any $n \in \mathbb{N}$ and any set of distinct indices $i_1,\ldots,i_n$ and any intervals $I_0,\ldots,I_n$ we have
$$P(N \in I_0, X_{i_1} \in I_1, \ldots, X_{i_n} \in I_n) = P(N\in I_0) P(X_{i_1} \in I_1, \ldots, X_{i_n} \in I_n)$$
3) for any $n \in N$ and any bounded continuous $f:\mathbb{R}\to\mathbb{R},g:\mathbb{R}^n \to \mathbb{R}$ we have
$$ E[f(N)g(X_1,\ldots,X_n)] = E[f(N)] E[g(X_1,\ldots,X_n)] $$
and there are many other equivalent conditions. Their equivalence and the most natural definition of "independent" is more the subject of a measure theory and so I will not go into detail.