Built a sequence

Question

They ask me to create a sequence $\{A_n\}$ that fulfill:

1. $P( \limsup A_n) = \alpha$ with $\alpha\in[0,1]$
2. $\sum_{n \geq 1}{P(A_n)} = \infty $ and $P( \limsup A_n) = 0$

For the first case I don't know how to start and do it, for the second my idea is something that the probabilities are equiprobable like $\frac{1}{n}$ because it doesn't converge but I don't know what $\Omega$ and probability take.

Answer 1

1)

Choose some event $A$ and let $A_n=A$ for every $n$.

Then $\limsup A_n=A$ so that $P(\limsup A_n)=P(A)\in[0,1]$.

If $\alpha$ some fixed number then you can go for $\Omega=\{0,1\}$, $A=\{0\}$, $P(\{0\})=\alpha$ and $P(\{1\})=1-\alpha$.

2)

You can take $\Omega=\mathbb N_+$ together with $P(\{n\})=\frac{1}{n}-\frac1{n+1}$ for $n=1,2,\dots$

Then for $A_n=\{n,n+1,n+2,\dots\}$ we have $P(A_n)=\frac1n$ and $\limsup A_n=\varnothing$ so that the conditions mentioned under 2) are satisfied.

Answer 2

For the first part you can take the probability space as $(0,1)$ with Lebesgue measure and $A_n=(0,\alpha)$ for all $n$.

For the second on elet $X$ be a random variable taking values $1,2,3...$ such that $P(X=n) =c/n^{2}$ where $c =\frac 6 {\pi^{2}}$. Let $A_n =(X>n)$. Then $\lim \sup A_n$ is the empty set and $\sum P(X>n)=\infty$.

[$\sum P(X>n)=\sum_n \sum\limits_{k=n+1}^{\infty} \frac c {n^{2}}$. Now $\sum\limits_{k=n+1}^{\infty} \frac c {n^{2}} \sim \frac c n$ and $\sum \frac 1 n =\infty$].

You can take $X$ to be any positive random variable with infinite expectation and the take $A_n=(X>n)$.