Integration of non-negative measurable functions

Question

Let $f$ be a non-negative measurable function whose domain is $X$. The the integral of $f$ is defined as -

$$\int_X f \mathrm d\mu = \sup\left\{\int_X s \mathrm d\mu: 0 \le s \le f\right\}$$

where $s$ is a measurable simple function.

I am trying to understand this. Is this saying that of the set $A$ of all possible simple functions where $0 \le s(x) \le f(x), s \in A, x \in X$, if we take the 'biggest' one it will fit $f$ infinitely closely, hence it's supremum will be $f$. Is my understanding correct here?

Answer 1

Yes you are right. Note that in general you may not able to find simple function that will coincide with $f$ on the set of full measure, and it may happen that integral for each such simple function will be strictly smaller than the final supremum.

Answer 2

Yes. The simple functions less than $f$, can be thought of as the generalization of lower Riemann sums in Riemann integrals. For a bounded Riemann integrable function $f:[0,1] \rightarrow \mathbb{R}$. Each lower sum is in fact a simple function and the supremum of the integrals of these simple functions gives the integral of $f$.