I have been working a lot on the following questions: find a sequence ($X_n$) of random variables such that $\mathbb{E}(\liminf X_n) = 0$ and $\liminf \mathbb{E}(X_n) = \infty$?
I think that the desired sequence should be such that $\exists k \in \mathbb{N}$ such that $X_n(\omega) = 0 \; \forall n\geq k$, while at the same time $\exists k' \in \mathbb{N}$ such that $\mathbb{E}[X_n(\omega)] = \infty \; \forall n\geq k'$, (but I am not sure whether I must pick $k \neq k'$ in order not to lose generality).
To find this example, I thought about defining a sequence of random variables $Y_m$ such that each of them is bounded, i.e. $\mid Y_m\mid < c_m$, with $c_m$ strictly increasing in $m$. The random variable $X_n = Y_1 + Y_2 + ... + Y_n$, so that it is the sum of the first $n$ variables $Y$'s, for each $n$. However, I get stuck when I try to formalize this. Is this the right way to provide the example I am looking for?
Thank you in advance.
A simple example: On the unit interval define $X_n$ to be a box of height $h_n$ over the interval $(0,\frac1n]$, zero otherwise. It's clear that $X_n$ tends to zero pointwise since the boxes are getting narrower and squeezing toward the origin. Therefore $\liminf X_n=0$. Meanwhile, under Lebesgue measure, the area of the $n$th box is $E(X_n)=h_n/n$. You can choose $h_n$ so that $E(X_n)$ meets any desired requirement.