For \(n\geq 1\) let \[ f_n \coloneqq n \chi_{[0, 1/n]}\colon\RR\to\RR. \]
For \(x<0\), \(f_n(x) = 0\) for all \(n\); for \(x>0\), \(f_n(x) = 0\) for all \(n > 1/x\). Hence \(f_n\to 0\) at every \(x\neq 0\), so \(f_n\to 0\) almost everywhere.
\(f_n(0) = n\to\infty\), so \((f_n)\) does not converge pointwise on \(\RR\), and in particular does not converge uniformly.
\(\int_\RR\abs{f_n - 0}\dx = 1\) for every \(n\), so \(f_n\not\to 0\) in \(L^1(\RR)\).
Give a sequence that converges a.e. but not in $L^1$, uniformly, or pointwise.