Continuous Partition Theory?

Question

Say I want to consider tuples $(x_1, \cdots, x_k)$ so that $$\sum_{i} x_i = c$$ for some constant $c$, with some additional restrictions on $x_i$. If $x_i$ are integers we have the partition theory that answers these sorts of questions. Is there an analogous field of study when $x_i$ are real? Obviously we can't count continuous partitions but we can integrate over partition domains or something like this.