In Royden, simple functions are defined to be functions that are real-valued, measurable, and finite-valued.
In chapter 4 Rodyen states that, given the subsets of real numbers $E_1, \dots, E_n$ and constants $a_1,\dots, a_n$,
$$\psi(x)=\sum_{i=1}^na_i\chi_{E_i}$$ is called simple if $E_i$ is measurable for each $i\in[n]$.
I am wondering is this an iff or if? I say this because if $E_i$ is measurable for each $i\in[n]$, then, yeah, $\psi$ is simple. However, $$\chi_{[0,1]}=\chi_{P}+\chi_{[0,1]-P}$$ where $P$ is a nonmeasureable set.
Is this use of simple a new definition?
As your example shows, this statement is only compatible with the old definition when you take "if" to mean literally "if", not iff. It's definitely not meant to be interpreted as an iff (if you interpret it that way, then actually no function would be "simple"). The intended "iff" definition is actually that $\psi$ is simple iff there exist measurable sets $E_i$ and constants $a_i$ such that $\psi=\sum_{i=1}^na_i\chi_{E_i}$. That is, the assumption that the $E_i$ are measurable should be stated at the beginning.
(Note that it is not entirely obvious that this is equivalent to the old definition of "simple" , since it is not entirely obvious that $\sum_{i=1}^na_i\chi_{E_i}$ is finite-valued. To prove that it is, observe that for any $x$, $\sum_{i=1}^n a_i\chi_{E_i}(x)=\sum_{i\in S}a_i$ where $S=\{i:x\in E_i\}$, and there are only finitely many different sets that $S$ could be.)