Lebesgue Integral for Non-negative Functions

The Lebesgue integral of a non-negative function $$f$$ is defined as:

$$\int f(x)\,dx=\sup_g \int g(x)\,dx,$$

where the supremum is taken over all measurable functions $$g$$ such that $$0\leq g\leq f$$, and $$g$$ is bounded and supported on a set of finite measure. (*)

My Question

1. When we take the supremum, do we assume the support of $$g$$ is all same for the measurable functions $$g$$?

2. In effect, is this what we are doing in the definition? Consider all measurable functions $$g$$ that satisfies (*). Consider their Lebesgue integration and pick the least upper bound of that set. This set is a set of numbers (e.g. Lebesgue measure) each corresponding to the Lebesgue integration number of each $$g$$.

3. Why does this definition make sense when we consider non-negative functions?

Reference: $$\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.

As you said we define for non-negative functions. For the first question is yes we consider an integral over a fix set $$\int_E f$$ we pick any $$g$$ bounded on $$E$$ not just this but there exists $$A\subset E$$ and $$m(A)<\infty$$ and $$g=0$$ on $$E-A$$ by this way we can connect it with the preceding build.