Egorov's Theorem - Counterexample in Infinite Case

Question

Why does Egorov's theorem not hold in the case of infinite measure? It turns out that, for example, $f_n = \chi_{[n,n+1]}x$ does not converge nearly uniformly, that is, it does not converge on E such that for a set F m(E\F) < $\epsilon$. Is this simply true because it takes on the value 1 for each n but suddenly hits 0 when n ---> infinity?

Answer 1

$f_n$ converges pointwise to the zero function on $\mathbb{R}$ (here $E = \mathbb{R})$. However there doesn't exist a set of finite measure $F$ such that $f_n$ converges uniformly on $\mathbb{R} \setminus F$. To see this note that for large enough $n$, $f_n$ will take both the values $0$ and $1$ on $\mathbb{R} \setminus F$.