This may be somewhat a repeat, but in this question, I'm asking for more of a conceptual answer rather than a proof. We define the Lebesgue integral, for a measurable function $f:X \longrightarrow [0,\infty]$ and a measurable set $E$, as $$\int_E f d \mu = \sup\big{\{}\int_E s d\mu : 0 \le s \le f, s \text{ is simple} \big{\}}.$$
This is also defined as the lower Lebesgue integral (similar to the lower Riemann integral). An upper Lebesgue integral as $$\inf \big{\{} \int_E s d \mu : f \le s, s\text{ is simple} \big{\}}.$$
To my understanding, these two don't always equal each other. If this is true, why do we only treat one of them as the proper Lebesgue integral? Do all of the theorems involving the upper integral hold with the lower integral definition? What is preferable about the given definition?