Let $(W, \mathcal{A}, P, T)$ be an ergodic, invertible dynamical system, where $P$ is a dynamical system, and let $n(w) = \inf \{ j \in \mathbb{N} : T^j w \in K \}$, where $K$ is an event with positive probability. Let $A \in \mathcal{A}$, and for some $w \in K$, set $$L_w = \{ w' \in K : n(w') = n(w) \textrm{ and } \chi_{A}(T^jw) = \chi_A(T^j w') \textrm{ for all $j = 0, 1, \ldots, n(w) - 1$} \} .$$ In other words, $L = L_w$ consists of all $w' \in K$ with the same return return time as $w$ and whose orbits visit $A$ at the same time as $w$'s does.
This paper I'm reading claims that if $P(L) > 0$, then $$\frac{P(A \cap L)}{P(L)} = \frac{1}{n} \sum_{j = 0}^{n - 1} \chi_A (T^j w) ,$$ where $n = n(w)$. I'm not sure how to prove this claim. My idea was that if I set $$g = \frac{1}{n} \sum_{j = 0}^{n - 1} \chi_A \circ T^j ,$$ then this is a random variable which is constant on $L$, so $g(w) = \frac{1}{P(L)} \int_L g \mathrm{d} P$. Thus it'd be enough to prove that $\int_L g \mathrm{d} P = P(A \cap L)$. Trouble is that I don't know how to show that. I've figured out that \begin{align*} \int_L g \mathrm{d} P & = \frac{1}{n} \sum_{j = 0}^{n - 1} P(L \cap T^{-j} A) , \end{align*} but I don't know how to turn this into the value I want, and am even concerned that I might be on the wrong track. I'd appreciate any help with how to show that $P(A \vert L) = g(w)$.
The claim is false. Indeed, if $w' \in L$, then $\chi_A(w') = \chi_A(w)$, so $\frac{P(A\cap L)}{P(L)} = \chi_A(w)$, and thus as long as $n(w) > 1$, we may take $A$ to be any measurable set containing $w$ but not $Tw$.