Simple Functions: L1-Cauchy?

Question

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?

Answer 1

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.)