It is defined as follows. Let Omega be a set
of real numbers, F= {all subsets of Omega}, let mu(A)= # of integers in A for any A in F, then
mu is a measure on (Omega,F) and mu is ``counting measure''.
Counting measure is a sigma-finite measure in the sense that there exists a countable partition of Omega,
Omega=\sum_{n=1}^{\infty} A_n, A_n in F such that \mu(A_n)<\infty for any n.
so if you let Omega be your R (real line), and note R= Union (i,i+1] for i = +/- 1, +/- 2 , ....
you will get your dream fulfilled!