If $m(A_k)\ge (\frac{5}{9})^k$, where $A_1, \dots, A_n$ in $(X,M,m)$, for which $n\, \exists A_i, A_j$ s.t. $A_i \cap A_j \neq \emptyset$?

Question

I am struggling to find a good idea to set up this exercise, could someone give me some hints about it? Thank you.

Let $(X,M,m)$ be a measurable space s.t. $m(X) = 1\,$, let be $n\ge2$ and $A_1, \dots , A_n$ measurable sets s.t. $m(A_k)\ge (\frac{5}{9})^k$, $k \in \{1,\dots,n\}$ . For which $n \in \mathbb{N}$ are we sure that $\exists A_i, A_j$, with $i \neq j$ s.t. $A_i \cap A_j \neq \emptyset$? If it was $m(A_k)\ge (\frac{1}{4})^k$ is it true that $A_i \cap A_j = \emptyset$, $\forall i, j$ s.t. $i \neq j$?

Answer 1

By the pigeonhole principle, $n$ just needs to be big enough so that $\sum_{k=1}^{n} (5/9)^k > 1$. In the second case, $\sum_{k=1}^\infty (1/4)^k < 1$ so there is no such $n$.