Let be $(X, \mathbb{A}, \mu)$ a measure space, a partition of $ X $ is a disjoint family $\xi=\{P_1,\ldots,P_k \}$ of measurable sets such tath $\bigcup P_i=X\pmod0).$
If $\xi=\{P_1,\ldots,P_k \}$ is a partition of $X$ or a cover of $X$ we define $$ \xi^n=\{P_{i_1}\cap T^{-1}P_{i_2}\cap\cdots \cap T^{-n+1}P_{i_{n-1}},~~0\leq i_j\leq k,~~0\leq j\leq n-1 \} $$
Edit: $T:X\to X$ is a measurabe tranformation and $P_{i_{j}}$ is any element of $\xi$.
My Problem:
Let be $\xi=\{P_1,\ldots,P_k \}$ a partition of $X$ and $\eta=\{P_1\cup P_0,\ldots, P_k\cup P_0 \}$ a cover of $X$. Supose that each element of $\eta^n$ intersect at most two elements of $\xi^n$, show that: $$ \operatorname{Card} \xi^n \leq 2^n\operatorname{Card}\eta^n $$
