# Does the following function converge to zero almost certainly?

Let $X_n(w) = nI_{(n-1,n)}(w)$ be a function on the real numbers with the Borel sigma algebra. Consider Lebesgue measure, denoted by $\mu$. Can this be a random variable? Does this converge to 0 almost surely?This is equivalent to showing $f_n(x) = nI_{(n-1,n)}(x) \rightarrow 0 \ a.e.[\mu]$. What about pointwise? How do you prove this?

We can view the sequence $(X_n)$ as a sequence of measurable functions. Since the intervals $(n-1,n)$ are pairwise disjoint, we have for a fixed $\omega$ that $X_n(\omega)=1$ for at most one $n$. Hence the sequence converges pointwise to $0$. In particular it converges almost everywhere for any measure (probability measure or not).