Editing Symmetric difference (section)

==Symmetric difference on measure spaces==
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are.

First consider a finite set ''S'' and the [[counting measure]] on subsets given by their size. Now consider two subsets of ''S'' and set their distance apart as the size of their symmetric difference. This distance is in fact a [[metric (mathematics)|metric]], which makes the [[power set]] on ''S'' a [[metric space]]. If ''S'' has ''n'' elements, then the distance from the [[empty set]] to ''S'' is ''n'', and this is the maximum distance for any pair of subsets.<ref>Claude Flament (1963) ''Applications of Graph Theory to Group Structure'', page 16, [[Prentice-Hall]] {{mr|id=0157785}}</ref>

Using the ideas of [[measure theory]], the separation of measurable sets can be defined to be the measure of their symmetric difference. If μ is a [[sigma-finite|σ-finite]] [[measure space|measure]] defined on a [[sigma-algebra|σ-algebra]] Σ, the function
:<math>d_\mu(X, Y) = \mu(X\, \Delta\,Y)</math>

is a [[pseudometric space|pseudometric]] on Σ. ''d<sub>μ</sub>'' becomes a [[metric space|metric]] if Σ is considered modulo the [[equivalence relation]] ''X'' ~ ''Y'' if and only if <math>\mu(X\, \Delta\,Y) = 0</math>. It is sometimes called [[Fréchet]]-[[Nikodym]] metric. The resulting metric space is [[separable space|separable]] if and only if [[L^2|L<sup>2</sup>(μ)]] is separable.

If <math>\mu(X), \mu(Y) < \infty</math>, we have: <math>|\mu(X) - \mu(Y)| \leq \mu(X\, \Delta\,Y)</math>. Indeed,
:<math>\begin{align}
  |\mu(X) - \mu(Y)|
    &=    \left|\left(\mu\left(X \setminus Y\right) + \mu\left(X \cap Y\right)\right) - \left(\mu\left(X \cap Y\right) + \mu\left(Y \setminus X\right)\right)\right| \\
    &=    \left|\mu\left(X \setminus Y\right) - \mu\left(Y \setminus X\right)\right| \\
    &\leq \left|\mu\left(X \setminus Y\right)\right| + \left|\mu\left(Y \setminus X\right)\right| \\
    &=    \mu\left(X \setminus Y\right) + \mu\left(Y \setminus X\right) \\
    &=    \mu\left(\left(X \setminus Y\right) \cup \left(Y \setminus X\right)\right) \\
    &=    \mu\left(X\,  \Delta \, Y\right)
\end{align}</math>

If <math>S = \left(\Omega, \mathcal{A},\mu\right)</math> is a measure space and <math>F, G \in \mathcal{A}</math> are measurable sets, then their symmetric difference is also measurable: <math>F  \Delta G \in \mathcal{A}</math>. One may define an equivalence relation on measurable sets by letting <math>F</math> and <math>G</math> 
be related if <math>\mu\left(F  \Delta G\right) = 0</math>. This relation is denoted <math>F = G\left[\mathcal{A}, \mu\right]</math>.

Given <math>\mathcal{D}, \mathcal{E} \subseteq \mathcal{A}</math>, one writes <math>\mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A}, \mu\right]</math> if to each <math>D\in\mathcal{D}</math> there's some <math>E \in \mathcal{E}</math> such that <math>D = E\left[\mathcal{A}, \mu\right]</math>. The relation "<math>\subseteq\left[\mathcal{A}, \mu\right]</math>" is a partial order on the family of subsets of <math>\mathcal{A}</math>.

We write <math>\mathcal{D} = \mathcal{E}\left[\mathcal{A}, \mu\right]</math> if <math>\mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A}, \mu\right]</math> and <math>\mathcal{E} \subseteq \mathcal{D}\left[\mathcal{A}, \mu\right]</math>. The relation "<math>= \left[\mathcal{A}, \mu\right]</math>" is an equivalence relationship between the subsets of <math>\mathcal{A}</math>.

The ''symmetric closure'' of <math>\mathcal{D}</math> is the collection of all <math>\mathcal{A}</math>-measurable sets that are <math>= \left[\mathcal{A}, \mu\right]</math> to some <math>D \in \mathcal{D}</math>. The symmetric closure of <math>\mathcal{D}</math> contains <math>\mathcal{D}</math>. If <math>\mathcal{D}</math> is a sub-<math>\sigma</math>-algebra of <math>\mathcal{A}</math>, so is the symmetric closure of <math>\mathcal{D}</math>.

<math>F = G\left[\mathcal{A}, \mu\right]</math> iff <math>\left|\mathbf{1}_F - \mathbf{1}_G\right| = 0</math> <math>\left[\mathcal{A}, \mu\right]</math> [[almost everywhere]].