I want to costruct a measure space $(X,\mathcal{F},\mu)$ and a $\mathcal{C}\subset\mathrm{m}\mathcal{F}$, where $\mathrm{m}\mathcal{F}$ be the set of extended real-valued measurable functions on $X$, with following properties:
- $\mathcal{C}\subset L^1 (X)$,
- for any $\epsilon>0$ there exists a $\delta>0$ such that for any $f\in\mathcal{C}$ and for any $A\in\mathcal{F}$ with $\mu (A)<\delta$, $\int_A |f|d\mu<\epsilon$ holds,
- there exists $\epsilon>0$ such that for any $K>0$, $\mu(|f|>K)\ge\epsilon$ holds for some $f\in\mathcal{C}$.
Any help?
The problems here can come from atoms. If $X$ has only one element, say $x_0$, say of measure one and $\mathcal C$ is the class of all the function from $X$ to $\mathbb R$ which map $x_0$ to some $c$, the first item is satisfied. The second as well, since $\delta=1/2$ is always a good choice. And also the third because we can choose $\varepsilon=1/2$.