On the definition of Lebesgue integral

Question

In Royden's 'Real Analysis', he first defines the Lebesgue integral of simple functions that vanish outside a set of finite measure and then extends this to non-negative bounded measurable functions that vanish outside a set of finite measure. But in Rudin's 'Real and Complex Analysis', he straightaway defines the integral of a non-negative function as the limit of integrals of simple functions. He does not include any finite measure support considerations. My question is, what is the difference between these two approaches?

Answer 1

I wanted to put it as a comment, but I didn't have enough place.

A characteristic function $1_A$ is integrable $\iff$ $m(A)<\infty $.

So, probably in Rudin, they suppose that a simple function is defined as $$\sum_{i=1}^n a_i 1_{A_i}$$ where $m(A_i)<\infty $, whereas in Royden, they define a simple function as $$\sum_{i=1}^n a_i1_{A_i}$$ without assumption that $A_i$ have finite measure.