Natural density and Poincare's recurrence Let $(X,\mathcal{B},\mu,T)$ be a dynamical system and let $A \in \mathcal{B}$ such that $\mu(A)>0$ and $\forall x \in X$ we define $$L_x=\{n \in \Bbb{N}|T^nx \in A\}$$
Prove that $\mu(\{x:\bar{d}(L_x)>0\})>0$ where $$\bar{d}(L_x)=\limsup_n \frac{|L_x \cap \{1,2...n\}|}{n}$$.
My first thought is that i have to use somewhere Poincare's recurrence theorem.
But i cannot understand the behavior of the density function.
From Poincare we just know that for almost every $x \in A$ we have that $L_x$ is infinite.
Can some give me a hint to solve this?
I clearly do not want a full solution. 
I want hint to solve this only with Poincare's Theorem and the properties of the natural  density function.(Not Ergodic Theorems)
Any help is appreciated.
Thank you in advance

For almost every $x$, the limit $$\lim_{N \to \infty} \frac{1}{N}\sum_{n=1}^{N} \chi_A(T^nx) = \lim_{N \to \infty} \frac{|L_x \cap [1,N]|}{N}$$ exists. Let $f^*(x)$ be the limit. It is a fact (see Birkhoff ergodic theorem - note this does apply to non-ergodic measure preserving systems) that $\int f^*(x)d\mu = \int \chi_A d\mu$. However, we only know this applies if $\mu(X)$ is finite (which I am assuming it is). Finish from here.
