Sigma-additive set function

Template:Short description Template:Mcn In mathematics, an additive set function is a function <math display=inline>\mu</math> mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, <math display=inline>\mu(A \cup B) = \mu(A) + \mu(B).</math> If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, <math display=inline>\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n).</math>

Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity.

The term modular set function is equivalent to additive set function; see modularity below.

Additive (or finitely additive) set functionsEdit

Let <math>\mu</math> be a set function defined on an algebra of sets <math>\scriptstyle\mathcal{A}</math> with values in <math>[-\infty, \infty]</math> (see the extended real number line). The function <math>\mu</math> is called Template:Visible anchor or Template:Visible anchor, if whenever <math>A</math> and <math>B</math> are disjoint sets in <math>\scriptstyle\mathcal{A},</math> then <math display=block>\mu(A \cup B) = \mu(A) + \mu(B).</math> A consequence of this is that an additive function cannot take both <math>- \infty</math> and <math>+ \infty</math> as values, for the expression <math>\infty - \infty</math> is undefined.

One can prove by mathematical induction that an additive function satisfies <math display=block>\mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu\left(A_n\right)</math> for any <math>A_1, A_2, \ldots, A_N</math> disjoint sets in <math display=inline>\mathcal{A}.</math>

σ-additive set functionsEdit

Suppose that <math>\scriptstyle\mathcal{A}</math> is a σ-algebra. If for every sequence <math>A_1, A_2, \ldots, A_n, \ldots</math> of pairwise disjoint sets in <math>\scriptstyle\mathcal{A},</math> <math display=block>\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n),</math> holds then <math>\mu</math> is said to be Template:Em or Template:Em. Every Template:Sigma-additive function is additive but not vice versa, as shown below.

τ-additive set functionsEdit

Suppose that in addition to a sigma algebra <math display=inline>\mathcal{A},</math> we have a topology <math>\tau.</math> If for every directed family of measurable open sets <math display=inline>\mathcal{G} \subseteq \mathcal{A} \cap \tau,</math> <math display=block>\mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G),</math> we say that <math>\mu</math> is <math>\tau</math>-additive. In particular, if <math>\mu</math> is inner regular (with respect to compact sets) then it is <math>\tau</math>-additive.<ref name=Fremlin>D. H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.</ref>

PropertiesEdit

Useful properties of an additive set function <math>\mu</math> include the following.

Value of empty setEdit

Either <math>\mu(\varnothing) = 0,</math> or <math>\mu</math> assigns <math>\infty</math> to all sets in its domain, or <math>\mu</math> assigns <math>- \infty</math> to all sets in its domain. Proof: additivity implies that for every set <math>A,</math> <math>\mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing)</math> (it's possible in the edge case of an empty domain that the only choice for <math>A</math> is the empty set itself, but that still works). If <math>\mu(\varnothing) \neq 0,</math> then this equality can be satisfied only by plus or minus infinity.

MonotonicityEdit

If <math>\mu</math> is non-negative and <math>A \subseteq B</math> then <math>\mu(A) \leq \mu(B).</math> That is, <math>\mu</math> is a Template:Visible anchor. Similarly, If <math>\mu</math> is non-positive and <math>A \subseteq B</math> then <math>\mu(A) \geq \mu(B).</math>

ModularityTemplate:AnchorEdit

Template:See also Template:See also

A set function <math>\mu</math> on a family of sets <math>\mathcal{S}</math> is called a Template:Visible anchor and a [[Valuation (geometry)|Template:Visible anchor]] if whenever <math>A,</math> <math>B,</math> <math>A\cup B,</math> and <math>A\cap B</math> are elements of <math>\mathcal{S},</math> then <math display=block> \phi(A\cup B)+ \phi(A\cap B) = \phi(A) + \phi(B)</math> The above property is called Template:Visible anchor and the argument below proves that additivity implies modularity.

Given <math>A</math> and <math>B,</math> <math>\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B).</math> Proof: write <math>A = (A \cap B) \cup (A \setminus B)</math> and <math>B = (A \cap B) \cup (B \setminus A)</math> and <math>A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A),</math> where all sets in the union are disjoint. Additivity implies that both sides of the equality equal <math>\mu(A \setminus B) + \mu(B \setminus A) + 2\mu(A \cap B).</math>

However, the related properties of submodularity and subadditivity are not equivalent to each other.

Note that modularity has a different and unrelated meaning in the context of complex functions; see modular form.

Set differenceEdit

If <math>A \subseteq B</math> and <math>\mu(B) - \mu(A)</math> is defined, then <math>\mu(B \setminus A) = \mu(B) - \mu(A).</math>

ExamplesEdit

An example of a Template:Sigma-additive function is the function <math>\mu</math> defined over the power set of the real numbers, such that <math display=block>\mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\

                              0 & \mbox{ if } 0 \notin A.

\end{cases}</math>

If <math>A_1, A_2, \ldots, A_n, \ldots</math> is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality <math display=block>\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)</math> holds.

See measure and signed measure for more examples of Template:Sigma-additive functions.

A charge is defined to be a finitely additive set function that maps <math>\varnothing</math> to <math>0.</math><ref>Template:Cite book</ref> (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range is a bounded subset of R.)

An additive function which is not σ-additiveEdit

An example of an additive function which is not σ-additive is obtained by considering <math>\mu</math>, defined over the Lebesgue sets of the real numbers <math>\R</math> by the formula <math display=block>\mu(A) = \lim_{k\to\infty} \frac{1}{k} \cdot \lambda(A \cap (0,k)),</math> where <math>\lambda</math> denotes the Lebesgue measure and <math>\lim</math> the Banach limit. It satisfies <math>0 \leq \mu(A) \leq 1</math> and if <math>\sup A < \infty</math> then <math>\mu(A) = 0.</math>

One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets <math display=block>A_n = [n,n + 1)</math> for <math>n = 0, 1, 2, \ldots</math> The union of these sets is the positive reals, and <math>\mu</math> applied to the union is then one, while <math>\mu</math> applied to any of the individual sets is zero, so the sum of <math>\mu(A_n)</math> is also zero, which proves the counterexample.

GeneralizationsEdit

One may define additive functions with values in any additive monoid (for example any group or more commonly a vector space). For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set. For example, spectral measures are sigma-additive functions with values in a Banach algebra. Another example, also from quantum mechanics, is the positive operator-valued measure.

ReferencesEdit

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