Let \(u \in \mathcal{H}^{1}\) and denote by \(\Pi_{n}, n \geqslant 1\), a sequence of partitions of \([0,1]\) such that \(\lim _{n ightarrow \infty}\left|\Pi_{n}ight|=0\). Show that the functions

\[f_{n}(t)=\sum_{t_{j}, t_{j-1} \in \Pi_{n}}\left[\frac{1}{t_{j}-t_{j-1}} \int_{t_{j-1}}^{t_{j}} u^{\prime}(s) d sight]^{2} \mathbb{1}_{\left[t_{j-1}, t_{j}ight)}(t), \quad n \geqslant 1\]

converge for \(n ightarrow \infty\) to \(\left[u^{\prime}(t)ight]^{2}\) for Lebesgue almost all \(t\).