Examples that are not Lebesgue integrable for any $p$

Question

I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but simple examples to me?

Answer 1

On $[0,1]$, try $$Z:=\sum_{j=1}^{+\infty}e^{e^j}\chi_{(2^{—j},2^{-(j+1)})}.$$

Answer 2

$\Omega = \mathbb{R}$ and $\mu((a,b)) = b-a$. $$Z = 1 ,\,\,\,\,\,\,\, \forall x \in \Omega$$ Then $\forall p \in \mathbb{R}$, $$\int_{\Omega} Z^p d \mu = \int_{\mathbb{R}} d\mu = \mu(\mathbb{R}) = \infty$$