Lebesgue Measure of an interval (a,b)

Question

I am following the book "Measure and Integration" by Leonard F. Richardson. In this book he develops the Lebesgue measure on $\mathbb{R}$ by :

1. Taking $X_n = [-N,N)$ and defining the measure on the field of elementary sets (by $\mu[a,b)= b-a$ ) and then using Hopf Extension theorem to extend this measure on the sigma algebra of Lebesgue Measurable sets of $X_N$.

2. Now using the $\sigma-$ finiteness of $\mathbb{R}$, we define Lebesgue measure on $\mathbb{R}$ by

$$l(A) = \sum l(A \cap S_N)$$ where $A$ is a subset of $\mathbb{R}$ , $S_N= X_N - X_{N-1}$, $N \in \mathbb{N}$.

Using this definition of measure of a subset of $\mathbb{R}$, how do we conclude that the measure of an open interval $(a,b)$ is $b-a$ ?

Please help.