Let $S$ be the set of all measures on some measurable space whose total variation is finite. Then this set is of course an Abelian group under pointwise addition and also a real vector space, which, if I'm not mistaken, is Banach with total variation as norm.
My question is: Is there a structure on (some restriction of) $S$ which turns it into an integral domain, or even a field, such that total variation becomes an absolute value (non-negative, positive-definite, subadditive and multiplicative)?