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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.2011-02-15
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    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?2011-02-15
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    P(R) is just a set. I'm not sure I understand. Do you want these functions to have any particular properties?2011-02-15

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