Let $X \sim$ Exp$(\lambda)$ and $Y:=\lfloor{X} \rfloor$.
How to prove that $Y$ is a random variable?
I used:
I want to show that $Y: \Omega \to \mathbb{R}$ is measurable.
Since $X \sim$ Exp$(\lambda)$, the cumulative distribution function is $F_X(x)=1-e^{-ay}$
I know that floor functions are monotone piecewise continuous functions.
So I wrote the floor function as
$$\sum_{n=-\infty}^\infty n \, \chi_{[n,n+1)}(x)$$
But how can I conclude that $Y$ is a random variable?