Question Regarding Lebesgue Integrable

I am reading Royden's Real Analysis(5th edition), and confusing about the definition of Lebesgue Integrable.

Here is the definition in chapter 4.2: A bounded function defined on a measurable set is said to be Lebesgue integrable or simply integrable, provided that$$\sup_{\varphi\leq f}\int_{E}\varphi dm=\inf_{f\leq\psi}\int_{E}\psi dm$$where the infimum and supremum are taken over finitely supported, simple functions on E. The common value of the infimum and supremum is called the Lebesgue integral, or simply the integral of f over E.

In chapter 4.5, it defines that: A measurable function $f:E\mapsto\bar{\mathbb{R}}$ is said to be integrable provided that $\int_{E}|f|<\infty$

I'm confusing that why don't we all use the second definition?