Simple functions are of the form $\phi(x) = \sum_{k=1}^N a_k \chi_{A_k}(x)$ where $\chi$ is the indicator function and that $A_k$'s are measurable sets. This is how Stein defines a simple function anyway.
Below are some thoughts and questions about the canonical form and the original form.
We need to place these simple functions in their canonical form, so that our $\bigcup_{k=1}^N A_k = \bigcup_{j=1}^M E_j$, but in such a way that the $E_j$ are disjoint. We complete the construction of the canonical form by noting that $\phi$ can only take on a set of finite values $\{c_1, \dots, c_k\}$, so we set $E_j = \{x: \phi(x) = c_j\}$.
My question revolves around the definition of the original simple function. Do we assume that a simple function has the traditional properties of a function, because the definition gives no indication as to whether something like this exists:
$$\phi(x) = \chi_{[0,1]}(x) + 2\chi_{[0,1]}(x) + 3\chi_{[0,1]}(x)$$
which is clearly not a function in the traditional sense, and it would prevent us from putting $\phi$ into a canonical form. It just seems to me like the definition of a simple function almost invites multiple values on intersecting sets, because there is no requirement like
$$E_k \cap E_j \neq \emptyset \implies a_k = a_j$$
Maybe this is overthinking, but I figure'd I'd ask just to check if I was missing something more important.
Best Answer
If it makes any easier, Simple functions are functions which take only fininitely many values. This is the simplest definition you'll get. This won't creat any confusion cause you know for any point $x $ in your domain, $f (x) $ is just one of the finitely many values. It is also easier to work with.
Try to see the equivalence of this with your definition . Hope it's helpful.