Sigma additivity - Citizendia

In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size (length, area, volume) of a set. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically

Formally, let μ be a function defined on an algebra of sets $\mathcal{A}$ with values in [−∞, +∞] (see the extended real number line). In Mathematics a field of sets is a pair \langle X \mathcal{F} \rangle where X is a set and \mathcal{F} is an algebra In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced The function μ is called additive if, whenever A and B are disjoint sets in $\mathcal{A},$ one has

$\mu(A \cup B) = \mu(A) + \mu(B) .$

(A consequence of this is that an additive function cannot take both −∞ and +∞ as values, for the expression ∞ − ∞ is undefined. In Mathematics, two sets are said to be disjoint if they have no element in common )

One can prove by mathematical induction that an additive function satisfies

$\mu(\bigcup_{n=1}^N A_n)=\sum_{n=1}^N \mu(A_n)$

for any A₁, A₂, . Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that . . , A_n disjoint sets in $\mathcal{A}$ .

Suppose $\mathcal{A}$ is a σ-algebra. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty If for any sequence A₁, A₂, . In Mathematics, a sequence is an ordered list of objects (or events . . , A_n, . . . of disjoint sets in $\mathcal{A}$ one has

$\mu(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu(A_n),$

we say that μ is countably additive or σ-additive.

Any σ-additive function is additive but not vice-versa, as shown below.

Useful properties of an additive function μ include the following:

μ(∅) = 0.
If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B).
If A ⊆ B, then μ(B - A) = μ(B) - μ(A).
Given A and B, μ(A ∪ B) + μ(A ∩ B) = μ(A) + μ(B).

Examples

An example of a σ-additive function is the function μ defined over the power set of the real numbers, such that

$\mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\ 0 & \mbox{ if } 0 \notin A. \end{cases}$

If A₁, A₂, . In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the real numbers may be described informally in several different ways . . , A_n, . . . 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

$\mu(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu(A_n)$

holds.

See measure and signed measure for more examples of σ-additive functions. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values

An example of an additive function which is not σ-additive is obtained by considering μ, defined over the power set of the real numbers by the slightly modified formula

$\mu (A)= \begin{cases} \infty & \mbox { if } 0 \in \bar A \\ 0 & \mbox { if } 0 \notin \bar A \end{cases}$

where the bar denotes the closure of a set. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the closure of a set S consists of all points which are intuitively "close to S "

One can check that this function is additive by using the property that the closure of a finite union of sets is the union of the closures of the sets, and looking at the cases when 0 is in the closure of any of those sets or not. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets That this function is not σ-additive follows by considering the sequence of disjoint sets

$A_n=\left[\frac {1}{n+1},\, \frac{1}{n}\right)$

for n=1, 2, 3, . . . The union of these sets is the interval (0, 1) whose closure is [0, 1] and μ applied to the union is then infinity, while μ applied to any of the individual sets is zero, so the sum of μ(A_n) is also zero, which proves the counterexample.

Another counterexample can be obtained similarly, defining μ again over the power set of the real numbers by

$\mu (A)= \begin{cases} 1 & \mbox { if } \exist a>0 \ s.t.\ (0,a) \subset A \\ 0 & \mbox { if } \not\exist a>0 \ s.t.\ (0,a) \subset A \end{cases}$

At first sight, this is the same as the previous example, with the exception of negative sets. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the real numbers may be described informally in several different ways However rather than requiring the closure to include 0, it is required that the set include an interval next to 0. This will mean that any two sets that have measure 1 must overlap for some interval (0, a), this makes the measure additive (easily) but not sigma additive, using the same example of sets as above.

One can prove that each set has measure 0 by taking a to be half of 1/(n+1), do the sum will still be 0, but the union will once again be (0, 1) which clearly satisfies the condition required to have measure 1.

Generalizations

One may define additive functions with values in any additive monoid (for example any group or more commonly a vector space). In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set. The limit of a sequence is one of the oldest concepts in Mathematical analysis. For example, spectral measures are sigma-additive functions with values in a Banach algebra. In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the Another example, also from quantum mechanics, is the positive operator-valued measure. In Functional analysis and quantum measurement theory, a POVM (Positive Operator Valued Measure is a measure whose values are non-negative Self-adjoint

Examples

Generalizations

See also