Justification of Lebesgue Integrability in the following statement

Question

In these notes I was reading, it is stated that for a measurable function $f:X\to [0,\infty]$, its Lebesgue Integral is $$ \int_E f \ d\mu =\sup\left\{\int_E s(x) \ d\mu\mid 0\le s\le f, s \ \text{simple}\right\}. $$ In this statement, are we just assuming that $f$ is Lebesgue Integrable? Would you not have to first verify that the upper and lower lebesgue integrals are equivalent? Does measurability imply Lebesgue Integrability? As I read in this question, the function $1/x$ is a counterexample to this statement, so I cannot see what justifies the statement. Is it just a definition?