I am a complete beginner in measure theory and have only been reading Tao's notes on them for the past couple of days. Therefore what I say might be a triviality or just plain wrong.
Anyway, Tao covers the Jordan measure, followed by the Lebesgue measure followed by the Lebesgue integral. His definition of the Lebesgue integral is stepwise, first defining it for simple functions and then approximating functions using these simple functions.
This approach is a little complicated, for instance you have to show that the decomposition into simple functions is irrelevant and so on. However, given the definition of a Lebesgue measure, would the following also be a definition of the Lebesgue integral:
For a function $f: \mathbb R^n \to \mathbb R$, define: $$\int fd\mu = \mu(\overline\Gamma_+) - \mu(\overline\Gamma_-)$$ where: $$\overline\Gamma_+ = \{(x,y) \in \mathbb R^n\times\mathbb R : y \in [0,f(x)], f(x) \geq 0\}$$ and similarly: $$\overline\Gamma_- = \{(x,y) \in \mathbb R^n\times\mathbb R : y \in [0,f(x)], f(x) \leq 0\}$$
We will call $f$ Lebesgue integral if $\overline\Gamma$ is Lebesgue measurable. If $\mu$ is the Jordan measure, then it is not hard to see that this is simply the usual definition of the Riemann integral and is in fact, quite close to how we normally think about it.
Is there any reason not to use this definition?