Formally, a simple function is a finite linear combination of indicator functions of measurable sets. More precisely, let $(X, Σ)$ be a measurable space. Let $A_1, ..., A_n \in Σ$ be a sequence of disjoint measurable sets, and let $a_1, ..., a_n$ be a sequence of real numbers. A simple function is a function $f:X\to\mathbb{R}$ of the form$$f=\sum^n_{i=1} a_i \space \chi_{A_i}.$$
If a measure $\mu$ is defined on the space $(X, Σ)$, the integral of $f$ with respect to $\mu$ is$$\sum^n_{i=1} a_i \space \mu(A_i)$$if all summands are finite.
This the definition of Lebesgue integral of a simple function taken from Wikipedia.
My question is: Why we need a restriction such that choosing $A_1, ..., A_n \in Σ$ be a sequence of DISJOINT measurable sets?
For example$$f_1=2 \space \chi_{[0,2]}+4 \space \chi_{[1,3]}$$and$$f_2=2 \space \chi_{[0,1)}+6 \space \chi_{[1,2]}+4 \space \chi_{(2,3]}$$are the same simple function. But according to definition given above we can not integrate $f_1$.
In some other sources, the definition is given as follows.
$f$ is a simple measurable function if and only if it is written as afinite linear combination of characteristic functions of measurablesets.
Lebesgue integral of a simple function is defined by$$\int_{X}fd\mu:=\sum^n_{i=1} a_i \space \mu(A_i).$$In this definition $A_i$ are not necessarily DISJOINT.
What is the difference between these two definitions?