I am trying to understand the intuitive derivation of the Frobenius-Perron (FP) operator in the monograph:
Lasota, Andrzej, and Michael C. Mackey. Chaos, fractals, and noise: stochastic aspects of dynamics. Vol. 97. Springer.
Specifically section 1.2. Assume we have a transformation $S :[0, 1] -> [0, 1]$ and pick a large number of initial states as $x^0_1, x^0_2,...,x^0_N$
Define an indicator function (or characteristic function) defined for a set $\Delta$ as
\begin{equation} 1_{\Delta}(x)= \begin{cases} 1 & if\,x\in\Delta\\ 0 & if\,x\notin\Delta \end{cases} \end{equation}
The we say that the function 𝑓𝑜(𝑥) is the density function for the initial states $x^0_1, x^0_2,...,x^0_N$ if for every (not too small) interval $Δ_o⊂[0,1]$ we have
\begin{equation} \int_{\Delta_o} f_o (u) du \approx \frac{1}{N} \Sigma_1 ^N 1_{\Delta_o}(x^0_j) \end{equation}
My conceptual difficulty is the right hand side of the above should be 1.
Unless the interpretation is that the initial states $x^0_j$ could be out of the $\Delta_o$. Please advise on my understanding of the definition of density of states. The text goes on to use this to give an intuitive derivation of the FP operator.
