Uniformly Integrability and Non-Tightness

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:

1. $$\mathcal{C}\subset L^1 (X)$$,
2. 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,
3. there exists $$\epsilon>0$$ such that for any $$K>0$$, $$\mu(|f|>K)\ge\epsilon$$ holds for some $$f\in\mathcal{C}$$.

Any help?

• What exactly do you want to ask? In the title you talk about non-tightness but 3) is tightness. – Kavi Rama Murthy Jun 12 at 5:46
• @KaviRamaMurthy Exactly. I fixed it. Thank you. – Ichiko Jun 12 at 5:54
• @saz Exactly. Some measure spaces have no such sets. So I want to costruct a suitable measure space $(X,\mathcal{F},\mu)$ and a $\mathcal{C}\subset\mathrm{m}\mathcal{F}$. – Ichiko Jun 12 at 6:49

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 $$\varepsilon=1/2$$.