Remark 1

Th. 6 no longer holds if the hypothesis $|f_n| \leq g$ (with $N_p(g) < +\infty$) is replaced by the weaker hypothesis $\sup_n N_p(f_n) < +\infty$. Suppose, for example, that $\mu$ is Lebesgue measure on $\mathbf{R}$; define continuous functions $f_n$ in the following manner: $f_n(x) = 0$ for $x \leq 0$ and for $x \geq \frac{2}{n}$, $f_n(\frac{1}{n}) = n$, $f_n$ being linear on the intervals $[0, \frac{1}{n}]$ and $[\frac{1}{n}, \frac{2}{n}]$. Then $\lim_{n \to \infty} f_n(x) = 0$ for all $x \in \mathbf{R}$, but $N_1(f_n) = 1$ for every $n$ (cf. §5, Exer. 12).