In a problem I was asked to to prove the following of a probabilistic model whose sample space is the real line:
$$P([0,\infty))=\lim_{n\to \infty}P([0,n])$$
The solution used the previously proved result $$P(A)=\lim_{n\to \infty}P(A_n)$$ where $A_n$ is an infinite sequence of events, with $A_n \subset A_{n+1}$ and $A=\cup_{n=1}^{\infty}A_n$.
Then the solution just set $A_n=[0,n]$ and $A=[0,\infty)$.
My question: Is it valid to compare and use somewhat interchangeably an interval and a set? Is it just possible in this case?