Definition of Lebesgue integral for simple functions

Question

I am reading lecture notes on Measure theory and Integration and I ran into the following definition and lemma which confuses me a lot.

Definition 1: Let $(X,\mathcal{M},\mu)$ be a $\sigma$-finite measure space. Let $f:X\to \mathbb{R}$ be a measurable function which takes on only finitely many values and any non-zero value is achieved on the set of finite measure. Then function $f(x)$ is called simple.

Definition 1': The function $f(x)$ is simple, if $$f(x)=\sum \limits_{k=1}^{n}c_k\chi_{E_k}(x),$$ where $E_k\in \mathcal{M}$, $E_k\cap E_j=\varnothing$ for $k\neq j$ and $\mu(E_k)<\infty$ for $c_k\neq 0$.

Remark: We note that any simple function can be written in the form $$f(x)=\sum \limits_{i=1}^{m}a_i\chi_{F_i}(x)$$ $a_1<a_2<\dots<a_m$ and $\sqcup_{i=1}^{m}F_i=X$. This is called canonical representation of simple function.

Definition 2: Let $f:X\to \mathbb{R}$ is a simple function and $$f(x)=\sum \limits_{i=1}^{m}a_i\chi_{F_i}(x)$$ is the canonical representation of $f(x)$. In this case we define Lebesgue integral of simple function $f(x)$ as follows: $$(L)\int \limits_{X}f(x)d\mu=\int \limits_{X}f(x)d\mu:=\sum \limits_{i=1}^{m}a_i\mu(F_i) \qquad \qquad(*)$$ (here we formally assume that $0\times \infty=0$).

Lemma 1(unchanged). The value of Lebesgue integral of simple function $f(x)$ does not depend on representation of $f(x)$ in the form $(*)$.

Answer 1

This is not really an answer, just some observations.

I think the presentation is a little confusing.

There are three characters in play.

Let me give my own name to the three things: (i) simple functions (Definition 1, finite range, finite support, the $E_k$ are disjoint), (ii) partition simple functions (the $E_k$ form a partition of $X$ and the $c_k$ need not be distinct) and (iii) 'ordered' canonical simple functions (as in the Remark, the $c_k$ are ordered and the $E_k$ are 'maximal').

(I have no idea why the author introduces the notion of ordered $c_k$.)

The author defines the integral in terms of (ii), the partition simple functions. Note that this representation is not unique. The $c_k$ may be repeated and hence the $E_k$ are not necessarily maximal.

For example, $1_{[0,2]} + 0\cdot 1_{[0,2]^c} = 1_{[0,1)} + 1_{[1,2]} + 0\cdot 1_{[0,2]^c}$ are two (partition simple function) representations of the same simple function. Obviously we have $1 \cdot m([0,2]) + 0 \cdot \infty = 1 \cdot m([0,1)) + 1 \cdot m([2,1]) + 0 \cdot \infty$, but the result still needs to be established formally so that Definition 2 makes sense.

So, a priori, the definition is ambiguous. Hence the author needs Lemma 1 to establish that any of the partition simple function representations gives the same value for the integral.

(As an aside, note that Lemma 1 is the first step in showing that the integral is linear.)

Personally I would prefer to define the integral in terms of a canonical representation (I am willing to forgo the ordering :-)) and then show that any other representation yields the same integral.