Let $(X,\mathcal{A}, \mu)$ be a measure space. Usually we define the Lebesgue integral for functions $X\to [0,\infty]$ which are measurable (i.e preimage of every Borel set in $[-\infty,\infty]$ lies in $\mathcal{A}$). Now, if $F,G:X\to [0,\infty]$ are measurable and $F=G$, $\mu$-a.e (i.e there is a $Z\in \mathcal{A}$ such that $\mu(Z)=0$ and $F=G$ on $Z^c$) then it is easy to show that $\int_X F\, d\mu = \int_X G \, d\mu$.
Consider the following definition:
Definition.
Suppose $f:X\to [0,\infty]$ is a function which is equal $\mu$-a.e to a measurable function $F:X\to [0,\infty]$. Then we define $\int_X f \, d\mu := \int_X F \, d\mu$, where on the RHS we have the standard Lebesgue integral of a non-negative measurable function.
This is certainly well-defined, because if $G$ is another such function then $F=G$, $\mu$-a.e, so that by my previous remark, this definition assigns the same value to $\int_X f \, d\mu$.
Questions:
Is making this definition ill-advised/not recommended for any reason?
Of course this becomes a non-issue if we're dealing with a complete measure space, so my question is whether we lose any generality by always working with the completion of a measure space? Are there any situations where we rather work with an incomplete measure space as opposed to its completion?
The context for these questions is that I've been studying the Bochner integral for Banach-space valued functions, and when seeing how this works out when the Banach space is $E\in \{\Bbb{R}, \Bbb{C}\}$, I noticed that there are quite a few additional technicalities regarding measure zero sets. This arises because a $\mu$-integrable function $f:X\to E$, i.e a function which is the $\mu$-a.e pointwise limit of a Cauchy-sequence (relative to $\mathcal{L}^1$-seminorm) of $\mu$-simple functions, is not necessarily measurable in the preimage sense.
$f$ is only equal $\mu$-a.e to a function $\tilde{f}:X\to E$ which is preimage-measurable, and this extra complication makes the statement of several theorems (which try to relate Bochner integrability and Lebesgue integrability, and the values of their integrals) kind of awkward.