I was reading this textbook Some Random Series of Functions by Jean Pierre Kahane. The attached scripts come from the page $9$ in the book.
My first question is why we have the equality in the red bracket?
My second question is, given a general random variable $X$ and a real number $r$, why we have $\mathscr{P}(X>r)$ is either $0$ or $1$?
I guess there must be some gaps which he omitted and I can't fill up. Thanks for help.