The problem that I am going to expose is linked to the interpretation of the definition of simple integrable function. The procedure used by the text is the standard one, before the simple functions are defined, in particular the non-negative ones, in the way: \begin{equation} s(x)=\sum_{i=1}^{n} c_i\chi_{A_i}(x), \end{equation} where $c_i\ge 0$ for $i=1,2,\ldots,n$ e $A_i=\{s=c_i\}$, $s(X)=\{c_1,\dots,c_n\}$.
The text does not provide the conditions of integrability, but, called $\lambda$ the measure of Lebesgue, defines the integral of $s$ in the following way:
\begin{equation} \int_X s(x)\;d\lambda=\sum_{i=1}^{n} c_i\lambda(A_i). \end{equation} Now, it is clear that as it is defined, the integral can assume the value $+\infty$, moreover, we use the convention that $0\cdot(+\infty)=0 $. The definition of simple function given leaves no room for interpretation, in the sense that a simple function, in general, can be written in many different ways, but the way of writing it above, called canonical form, is unique. In addition to text I have been provided with lecture notes in which a simple function is any function that can be written as \begin{equation} s(x)=\sum_{i=1}^{n} c_i\chi_{A_i}(x), \end{equation} where $c_i\in\mathbb{R}$ and $A_i$ are measurable together. Moreover, without affecting the generality, we can always assume that $c_i\ne 0$ and that the $A_i$ are disjoint, in practice, it is as if we "adjusted" the image of the function. What puzzles me is the definition of integrability: $s$ is integrable if $\lambda(\cup_{i = 1}^n A_i)<+\infty$ in this case arises: \begin{equation} \int_X s(x)\;d\lambda=\sum_{i=1}^{n} c_i\lambda(A_i). \end{equation} It seems to me that this last definition is less general than the first, because to be integrable a simple function $s$ must either be defined on a finite set or must be worth $0$ on unlimited sets (right?), because example if \begin{equation} s_1(x)= \begin{cases} 1 & \text{if $x\in [0,1]$} \\ 2 & \text{if $x\in(1,2]$} \\ 0 & \text{if $x\in(2,+\infty)$} \end{cases} \end{equation} According to the second definition $s_1 (x)$ is not integrable, but since we have to make it integral we can not consider the part where it is worth $0$ and consider \begin{equation} s_1(x)= \begin{cases} 1 & \text{if $x\in [0,1]$} \\ 2 & \text{if $x\in(1,2]$} \end{cases} \end{equation} In this way it is integrable and the integral made with the two definitions coincides, even if with the first one we could have proceeded without removing this part due to the convention.
I ask only if these considerations are correct and if there are other considerations to be made on the two definitions,
Thank you for the time you dedicate to me.