Functions on P(R) - are there examples? What are some examples of functions on the Power Set of the Reals? Is this an abuse of terminology - functions on the reals can be thought of as functions on the power set of the naturals with a specific ordering. I was hoping someone would kindly refer me to a text or article where explicit (not necessarily 'useful') examples of functions with the domain P(R) are given; or if this is confused idea why there is nothing to it. Thanks!
 A: A function on $\mathcal{P}(\mathbb{R})$ essentially means a "rule" of assigning to each subset of $\mathbb{R}$ an element of some set $S$, the codomain.  So e.g., there is the identity function $\mathcal{P}(\mathbb{R})\to\mathcal{P}(\mathbb{R})$ that sends a set $A$ to itself.  There is the $\sup$ map from $\mathcal{P}(\mathbb{R})$ to the extended reals $[-\infty,\infty]$ that sends $A\subseteq\mathbb{R}$ to $\sup A$, and similarly with $\inf$.  You could also define functions like $f(A)=1$ if $A$ is open and $f(A)=0$ if $A$ is not open, or other such maps indicating topological properties of subsets of $\mathbb{R}$.  You could define the function $c:\mathcal{P}(\mathbb{R})\to \{0,1,2,\ldots,2^{\aleph_0}\}$ such that $c(A)$ is the cardinality of $A$.  Or $C:\mathcal{P}(\mathbb{R})\to \{0,1,2,\ldots,2^{\aleph_0}\}$ such that $C(A)$ is the cardinality of the set of connected components of $A$.
I see that Zhen Lin has indicated a couple of other useful examples in the comments.  The complement in $\mathbb{R}$ defines a bijection on $\mathcal{P}(\mathbb{R})$.  Each outer measure on $\mathcal{P}(\mathbb{R})$ defines a function from $\mathcal{P}(\mathbb{R})$ to $[0,\infty]$.  The closure and interior maps are other functions from $\mathcal{P}(\mathbb{R})$ to itself, mapping onto the set of closed and open subsets of $\mathbb{R}$ respectively.
So yes, there are lots of explicit examples, but I don't know exactly what you're looking for.  The set of functions from $\mathcal{P}(\mathbb{R})$ to any fixed set $S$ is $S^{\mathcal{P}(\mathbb{R})}$, with cardinality $\displaystyle{|S|^{2^\mathfrak{c}}}$, which is at least $\displaystyle{2^{2^\mathfrak{c}}=2^{2^{2^{\aleph_0}}}}$ if $S$ has more than $1$ element.
A: Every set can be a domain of a function (in fact, a proper class can also be the domain of a function if one is careful enough). In particular the powerset of the real numbers can be.
Some examples that might be used are measures in measure theory, granted the Lebesgue Measure is not defined for every subset of the real numbers (unless the axiom of choice is not assumed) but you can define the outer measure which is defined on $P(\mathbb{R})$, or some other measure that is not limited by the requirements of the Borel/Lebesgue measures.
Other examples are negation, union and intersection (functions in two variables), as well symmetric difference.
Assuming the axiom of choice, the Stone-Cech compactification of the natural numbers with the discrete topology is of cardinality $\beth_2$ (i.e. $|P(\mathbb{R})|$) and you can look at it as if you are assigning each subset of the real numbers an ultrafilter over the natural numbers.
