# On the definition of Lebesgue integral

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?

• It is not necessary to consider the special case first. You c an define integrals for all non-negative simple functions and extend it to all non-negative measurable functions without worrying about finiteness of the integrals. Rudin's approach is more standard. Commented Nov 27, 2019 at 12:04

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.
• Wrong. Rudin does not require $m(A_i) <\infty$. See my comment above. Commented Nov 27, 2019 at 12:04