Simple Functions: L1-Cauchy? Given a measure space $(\Omega,\mathcal B(\Omega), \mu)$.
Consider simple functions:
$$s\in\mathcal{B}(\Omega,\mathbb{C}):\quad s=\sum_ka_k\chi_{A_k}$$
Suppose it is I-Cauchy:
$$\left|\int_\Omega(s_m-s_n)\mathrm{d}\mu\right|\stackrel{m,n\to\infty}\longrightarrow0$$
Does it imply L1-Cauchy:
$$\int_\Omega|s_m-s_n|\mathrm{d}\mu\stackrel{m,n\to\infty}\longrightarrow0$$
What example could serve?

Meanwhile I got it:
Given the real line $\mathbb{R}$.
Consider simple functions:
$$s_n:=-\chi_{(-n-1,-n]}+\chi_{(n,n+1]}$$
Then it is I-cauchy:
$$\left|\int_\mathbb{R}(s_m-s_n)\mathrm{d}\lambda\right|=0\stackrel{m,n\to\infty}{\to}0$$
But it is not L1-cauchy:
$$\int_\mathbb{R}|s_m-s_n|\mathrm{d}\lambda=4\stackrel{m,n\to\infty}{\to}0$$
(This example can be modified.)
