Sum Over a Field: Algebra, Group, Ring

[joebohr]

I remember encountering an instance when someone suggested that you could sum over a field other than the integers. I don't remember the exact topic, but I know someone here said it. How would one sum over a field, group, or ring? For that matter, how would you take an integral over a field, group, or ring (I don't mean field as in scalar or vector field, but rather in the sense of abstract algebra).

I wasn't sure whether to put this in the abstract algebra or analysis thread, but decided on analysis.

 

[mathman]

To take integrals you need to define a measure on the group, ring, or field.

 

[joebohr]

So if I define a measure μ on M, then would the integral be defined as \int_{μ}f(x)dx=\sum^{\infty}_{i=1}μ(A_{i})f(x) where M=\bigcup^{\infty}_{k=1}A_{k}? Or would it look like something different? How would I express this integral generally for a measure μ?

 

[mathman]

http://en.wikipedia.org/wiki/Haar_measure

Above is one example of how it could be done.

 

[joebohr]

http://en.wikipedia.org/wiki/Haar_measure

Above is one example of how it could be done.

Ok, that's a great example for an integral, but how would you find the sum over a field/group/ring?

 

[mathman]

Ok, that's a great example for an integral, but how would you find the sum over a field/group/ring?

I am not sure what you are looking for. For a sum you need to have weights assigned to each member of the group, ring, or field. It really doesn't matter whether or not there is an algebraic structure, only weights for each element.

 

[joebohr]

So it's impossible to solve a problem such as:

Sum the following expression over the field of rational numbers

Ʃ 1/x

where the sum over x ranges over all of the rational numbers between 1 and 3?

If this won't work with rationals because they are infinite, can someone come up with a structure in which this sum is possible (besides the usual ring of integers, of course).

Also, if this problem were an integral over the rationals instead of a sum, how would I use Haar measures to evaluate it? I looked at the wikipedia article but it doesn't seem to be very clear.

 

[mathman]

Since the integers are countable, your problem can't be done using Haar measure, since every number would have the same measure.

To do anything at all you would need to assign measures to each point. Your sum would look like: Ʃ m(x)/x, where m(x) is the measure of the point x.