It is defined as follows. Let Omega be a set of all positive integers, 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!