# 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!

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I'll assume you know that a function doesn't have to be defined in terms of explicit formulae or be necessarily number-valued or such. So here's a natural example: the complement operator is a function $\mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ taking $X \subseteq \mathbb{R}$ to $\mathbb{R} \setminus X$. Somewhat more useful examples of real-valued functions on the powerset may be found in measure theory, e.g. outer measures. –  Zhen Lin Feb 15 '11 at 18:08
Thank you kindly! Does it follow that all of the interesting real valued functions on P(R), or for that matter P(P(R)) are composite functions that begin with such a measure? –  mxyzptlk Feb 15 '11 at 18:25
P(R) is just a set. I'm not sure I understand. Do you want these functions to have any particular properties? –  Qiaochu Yuan Feb 15 '11 at 18:46

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.

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Thank you! I think my naivete is showing. I was thinking more along the lines of the functions from an intermediate calculus students toolbox. Inasmuch as polynomial functions, trigonometric functions are intuitively clear as functions on the real line or the complex plane, in a sense I was wondering - what are the functions that are interesting on the Power Set of the Reals, or that set's Power Set. Thank you for responding. –  mxyzptlk Feb 15 '11 at 18:48
As you can see, there are some examples that come up in analysis. Of these, I would say that outer measures are the ones most likely to be thought of explicitly as functions defined on the power set (even though $\sup$ and $\inf$ can also be thought of in this way, for example). Some commonly used outer measures include Lebesgue-Stieltjes outer measures and Hausdorff outer measures. –  Jonas Meyer Feb 15 '11 at 18:54
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.
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.