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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.

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First we define the integral for simple functions, then we define the integral for bounded functions on bounded sets! in a similar way to the Riemann integrals.

Now we move two steps we define integrals of unbounded functions on unbounded sets by

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.

For the second and the third I think should be clear by above.

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