Why don't we have "upper and lower" Lebesgue integrals?

Question

For a function to be Riemann integrable, the upper and lower Riemann sum need to be equal. However, this no longer applies to Lebesgue integrals. Let $(\Omega,\Sigma,\mu)$ be a measure space and define for $f\in \Sigma$ that $$ \text{SF}(f)=\{s\in \Sigma: s\text{ is a simple function and } 0\leq s(x)\leq f(x), \forall x\in X\}\\ \text{SG}(f)=\{s\in \Sigma: s\text{ is a simple function and } s(x)\geq f(x), \forall x\in X\} $$ Here is my question: why are we defining $$ \int_X f d\mu=\sup_{s\in SF(f)} \int_X sd\mu$$ rather than $$ \int_X f d\mu=\inf_{s\in SG(f)} \int_X sd\mu??? $$ Of course, the later definition force $f$ to be bounded, but why aren't we using the later definition?