Sequence of random variable converges to zero but expectation converges to infinity

Question

I am currently working on a problem from a Probability Theory course:

Construct a sequence of random variables $\xi_n$ such that $\xi_n(\omega) \rightarrow 0$ for every $\omega$, but $\mathbb{E} \xi \rightarrow \infty$ as $n \rightarrow 0$.

I find really hard to construct such a sequence given that I don't even know the probability measure of each $\omega$. Does each $\xi_n$ have to be a constant function?

Answer 1

Let $X_n(t)=n^{2}I_{(0,\frac 1 n)}$ on the space $(0,1)$ with Lebesgue measure. Then $X_n(t) \to 0$ for every $t$ and $EX_n=n$.