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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?


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closed as off topic by Dan Petersen, Mark Meckes, Kevin O'Bryant, Bill Johnson, Did Jan 29 '12 at 19:53

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Incomprehensible homework, voting to close. – Igor Rivin Jan 11 '12 at 13:44
To avoid closure, you need to rewrite it more carefully. But probably it is not a "research level" question even so, and should be at another site, not here. – Gerald Edgar Jan 11 '12 at 13:50

It may help to consider the indicators of the intervals [0,n] on the probability space [0,1] with Lebesgue measure.

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