$\newcommand{\scrF}{\mathscr{F}}$
I'm trying to undertstand the difference between the standard vs. a regular representation for a simple function. The definitions I'm working with are as follows.
Definition: Let $(\Omega, \scrF)$ be a measurable space and let $\phi:\Omega \rightarrow [0,\infty)$ be a simple function; this means that $\phi$ takes on finitely many values i.e , $\phi(\Omega) = \{a_1,a_2,\dots,a_m\}$. With this if we set $D_k = \{\omega \in \Omega \mid \phi(\omega) = a_k\} = \phi^{-1}[\{a_k\}]$, then we may write $$ \phi(\omega)= \sum_{k=1}^m a_k\chi_{D_k}(\omega) $$ where $\chi_{D_k}$ denotes the characteristic function of $D_k$ so $\chi_{D_k} = 1$ if $\omega \in D_k$ and $0$ otherwise. This is the standard representation of the simple function $\phi$, it always exists and is unique.
Definition: Let $(\Omega, \scrF)$ be a measurable space and let $\phi:\Omega \rightarrow [0,\infty)$ be a simple function. We define the more general regular representation of $\phi$ to be $$ \phi(\omega) = \sum_{k=1}^m a_k \chi_{D_k}(\omega), $$ where the $D_k$ are disjoint measurable sets such that $\cup_{n=1}^\infty D_k = \Omega$ but the $a_k$ are no longer required to be distinct.
My issue is what distinguishes these representations? Because in the definition of the standard representation of $\phi$ the sets $D_k$ are certainly disjoint since the $a_k$ are distinct, moreover since they're preimages of the entire range of $\phi$ it follows that $\cup_{k=1}^n D_k = \Omega$. Is the difference that in the regular representation of $\phi$ the sets $D_k$ no longer represent preimages of $\phi$ but are simply just measurable sets in $\scrF$?
Moreover, I don't see why the regular representation should even give $\phi(\omega)$, why can't it be the case that $\phi(\omega) = a_1$ but the regular representation contains no $a_1$ coefficient since they no longer need to be distinct and hence you no longer need to take all of the values in $\phi(\Omega) = \{a_1,\dots, a_m\}$. Thanks in advance for the clarification.
Edit: I think that I misunderstood the definition of a regular representation.
Definition revised: Given some simple function $\phi:\Omega \rightarrow [0,\infty)$, a representation $$ \phi = \sum_{k=1}^n a_k\chi_{D_k} $$ is called regular if the $D_k$ are disjoint measurable sets for which $\cup_1^n D_k = \Omega$ and $a_k\geq 0$ but not necessarily distinct.
So in this sense the standard representation for $\phi$ is a regular representation. Does this make sense?