I am a bit rusty on probability, though I did learn its basics in the axiomatic measure theoretic framework. So excuse me if the following is a trivial matter.
Let us start from a probability space $(\mathcal{U},\mathbb{P})$, where I suppose the implicit $\sigma$-algebra is the trivial one, ie the powerset $\wp(\mathcal{U})$.
Define a sample to be a (finite) sequence of elements $s_1,...,s_n\in\mathcal{U}$, whose order we forget.
The sample defines a new probability measure on $\mathcal{U}$, namely $$\forall A\subset\mathcal{U}\quad Fr(s,A)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}_{s_i\in A}$$
Nowm, fixed $A\subset\mathcal{U},\epsilon\in\mathbb{R}$, it does make sense to consider the following quantity $$\mathbb{P}\left(\{s_1,...,s_n\}\subset\mathcal{U}\ st\quad |Fr(s,A)-\mathbb{P}(A)|>\epsilon \right)$$
This is the measure of size $n$ samples which do not appproximate $\mathbb{P}$ on $A$ up to an $\epsilon$ value.
At the same time, we can define a random element $S$ of $(\mathcal{U},\mathbb{P})$ to be a $\mathcal{U}$-valued random variable such that: $$\forall A\subset\mathcal{U}\quad \mathbb{P}(A)=\mathbb{P}(S\in A)$$
First of all: formally, which is the probability measure on the rhs?
Then, we can also extend the notion and consider a random sample $S_1,...,S_n$.
Now, fixed an $A\subset\mathcal{U}$, to each of these elements we can associate a Bernoulli, valued in $\{0,1\}$ which is simply $\mathbb{I}_{S_i\in A}\sim Bern(\mathbb{P}(A))$.
Then we can mimik the defintion of frequency above to obtain a $1/n$-scaled Binomial(n,$\mathbb{P}(A))$
$$Fr(S,A)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}_{S_i\in A}$$
Then it seems to me that the following equality is often exploited:
$$\mathbb{P}\left(\{s_1,...,s_n\}\subset\mathcal{U}\ st\quad |Fr(s,A)-\mathbb{P}(A)|>\epsilon \right)=\mathbb{P}(|F(S,A)-\mathbb{P}(A)|>\epsilon)$$
Now, which is formally, the probability measure on rhs?
Why is the equality true?
Thanks in advance