Let $f_n = χ_{[n,2n]}$. Show that
$$\lim_{n \to \infty } f_n(x) = 0$$
for each $x ∈ R $ but
$$\lim_{n \to \infty } \int_R f_n \;\mathrm{dμ} \neq 0 = \int_R 0 \;\mathrm{dμ}$$
Also could you tell me why does this does not conflict with the monotone convergence theorem?
It should be obvious that each function has non-zero integral, and specifically has integral $n$. Thus the limit of the integrals diverges and specifically isn't equal to $0$.
However, when you take the limit of the functions you get something uniformly $0$. To see this, notice that for every $x\in\mathbb{R},\forall n>|x|$, $f_n(x)=0$ holds. Thus the limit function is uniformity zero and so has zero integral.
This doesn't contradict the monotone convergence theorem because it's not monotone.
