I am thinking about an integral of an indicator function. The integral could be as follows:
$$\int_W 1\{x\in W\}dN$$
I am thinking of this in the context of a point process $\{x_i\}$ on the real line. Can I think of this integral as representing the number of points in window $W$, i.e.
$$\int_W 1\{x\in W\}dN=\sum_i 1\{x_i\in W\}$$
If not, does there exist a function $f$ such that $$\int_W f dN=\sum_i 1\{x_i\in W\}$$
Thank you in advance, and please let me know if I can provide any further clarifying information or if my question is poorly specified.