Division of Characteristic Functions

[BSMSMSTMSPHD]

I'm in graduate analysis this year, but I've been out of school as a teacher for 6 years so I'm a bit rusty. Any help would be appreciated on this simple question. My apologies for not knowing the Math-type.

Yesterday in class we were discussing an example which demonstrated the Radon-Nikodym theorem. The results aren't important, but what is (to me) is that part of the result required the division of two characteristic functions defined on closed sets on the real line: (X[3,5]/X[3,7]). The result of this division was X[3,5], which I believed to be true.

Now, earlier this year, we showed that for any measurable space (Y,M), the product of characteristic functions is the characteristic function of the intersection. (ie: X(E)X(F) = X(E and F), for all E,F in M) I also believe this to be true.

So, how is X(E)/X(F) defined - in general? One other student I talk with suggested (like in the example) that the result is the same (ie: X(E)/X(F) = X(E and F)) but this is only true (IMO) if E is a subset of F.

So - my question then... How IS division of characteristic functions defined?? Is it always defined? Is it possible to write it as "multiplication of the complement?" What is X[1,5]/X[3,7] (for Y = [1,7] for example)?

Thanks for your help!

 

[BSMSMSTMSPHD]

By the way, in case it wasn't obvious, my notation X[a,b] means the function that equals 1 on the interval [a,b] and 0 everywhere else.

 

[mathman]

Mt math education goes back many years, so it is possible there has been some new definition. However, any arithmetic operation involving two functions was defined simply as the function resulting from the arithmetic operation being performed on every point in the domain. This applied to characteristic functions as well as any other. However, I never ran across a situation where characteristic functions were divided.