I am reading Folland's real analysis text, section 2.2 on integration of nonnegative functions. I am stuck at the definition of the integral of a simple function and how to show it is well-defined. There are related questions/answers to mine which I've included at the end together with a short comment.
Folland defines a simple function $f$ as a finite linear combination of characteristic functions. In its (unique) standard representation, a simple function has a distinct range $\{z_1,\ldots,z_n\},z_i\neq z_j,$ and the disjoint sets of the characteristic functions are simply $E_k=f^{-1}(\{z_k\})$.
Then, given $f=\sum_1^n z_j\chi_{E_j}$ in its standard representation, the integral of $f$ with respect to $\mu$ is $$\int f\,d\mu=\sum_1^n z_j\mu(E_j).$$This definition is followed by a proposition on basic properties of the integral of a simple function, among them linearity.
Now, comparing with another text (Stein and Shakarchi's real analysis text), they use the very same definitions (except that Folland allows one of the coefficients $z_i$ to be $0$). In their Proposition 1.1 on page 50, they first prove that the integral of a simple function is well-defined (i.e. independent of any representation) and then linearity. And proving linearity follows easily from the fact that it is independent of representation.
My question is; since Folland does not mention or comment on independence of representation, is it possible to deduce this from the linearity of the integral? In other words, doing the opposite of what Stein and Shakarchi are doing? I have not been able to understand this for days now and would be grateful for any clarification.
Related questions/answers:
- question 1 (asking why we need to prove independence of representation given the definition in Folland's text, but not really addressing how we can prove it given the definitions in Folland's text)
- answer 2 (the first part mentions linearity, yet I don't see how independence of representation follows from it)