Applying a function on any multiset of real numbers

Question

I would like to define a function whose domain is any multiset of real numbers and image is a real number.

To my understanding, the domain of a function that can be applied on any set of real numbers is the power set $\mathcal{P}(\mathbb{R})$. Is it correct? If yes, is there an equivalent "power multiset" and what is its notation?

Answer 1

I think you'd probably be best served by defining your own notation. You can write multisets as sets using various conventions (see Wikipedia page), and so you could perhaps view the function as having domain $\mathcal{P}(\mathbb{R}) \times \mathcal{P}(\mathbb{R}^2)$ (I believe). In the end though, it seems to me that you'd probably be better off with your own notation.

Answer 2

In the article Mathematics of Multisets by A. Syropoulos, this is defined as follows:

Assume that $A$ is a set, then $\mathcal{P}^A$ is the set of all multisets which have $A$ as their support set.