 Moving-spike obstruction to interchanging a limit and an infinite sum

Let $f(t,n)=1_{[n,n+1)}(t)$ for $t\ge 1$. For every fixed $n$, $f(t,n)\to 0$ as $t\to \infty$, but ${\sum }_{n}f(t,n)=1$ for every $t\ge 1$. Pointwise convergence without monotonicity or domination therefore does not justify interchanging a limit and an infinite sum.