Can partition be defined for numbers other than positive integers?

Question

An integer partition is a decomposition of a positive integer into positive integers that add up the original number. Apart from $\mathbb N_{> 0}$, does this definition apply to other numeric sets, like $\mathbb R$ or $\mathbb Q$?

Answer 1

I suppose you could define partitions on $\mathbb{R}$ (and $\mathbb{Q}$) similarly as you do on the positive integers. So say given a positive real number $r$, a partitioning of $r$ over $\mathbb{R}$ would be a (probably finite) collection of positive real numbers that sum to $r$. This wouldn't be as interesting, though, because given any real number, there are infinitely many ways to partition it over the reals.

What could be interesting would be to specify some weird $A \subset \mathbb{R}$ and talk about partitioning an arbitrary real number over $A$. Unfortunately, I can't think of a choice for $A$ that would yield anything nontrivial that is more interesting than partitioning over $\mathbb{R}$.