If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.

My first idea was

$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \leq E(Z^2)P(Z>\varepsilon) +\varepsilon P(Z\leq\varepsilon)$,

for some $\varepsilon > 0$.

As you can see, it's required that $E(Z^2)<\infty$ and it don't seems like an appropriate condition.

So my philosophical question is: Can we give to $E(Z)$ any sense?

Regards.