Definition talk:Simple Function

What's the benefit of providing the same definition for both real-number and non-real-number domains? From what I can tell, the definition is identical, even down to the difference that one invokes the term "lebesgue measurable"? (As even "lebesgue measurable" is exactly the same thing as "measurable" except that its domain is the set of real numbers.) --prime mover 02:21, 23 March 2012 (EDT)
 * I wanted to provide a more general definition of a simple function, but I wasn't sure whether I should keep or delete the real function version. Any suggestions on what to do here? –Abcxyz (talk | contribs) 09:13, 23 March 2012 (EDT)


 * As long as it is established somewhere that the real number line (under whatever conditions, it's still over my head yet) is a measure space, then the general definition should be sufficient as it stands. When it is necessary to invoke the definition in the real domain, it's at that point you say that "... the reals are a measure space, so this definition is valid" or whatever.


 * The way I see it is that this is a different situation from providing real-domain-only and complex-domain-only versions of general Analysis results, as (from what I understand) measure spaces are not generally studied in a specific domain. So I believe that there should be no need to maintain real-only versions of the proofs in this section.


 * I could be completely wrong, of course. Let me know if so. --prime mover 09:26, 23 March 2012 (EDT)


 * On the 'measure-spaceness' of $\R^n$, see Existence and Uniqueness of Lebesgue Measure which I happen to have posted just today. --Lord_Farin 09:36, 23 March 2012 (EDT)


 * There is a reason for simple functions to be considered (for now) only with real codomain; they are used to define Lebesgue integration. The only generalisation I have seen is to complex measures, replacing every $\R$ with $\C$. --Lord_Farin 15:10, 4 April 2012 (EDT)