- Sigma additivity
Additive (or finitely additive) set functions
(A consequence of this is that an additive function cannot take both −∞ and +∞ as values, for the expression ∞ − ∞ is undefined.)
One can prove by mathematical induction that an additive function satisfies
for any disjoint sets in .
σ-additive set functions
Suppose is a σ-algebra. If for any sequence of disjoint sets in one has
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).
If 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
where the bar denotes the closure of a set.
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. That this function is not σ-additive follows by considering the sequence of disjoint sets
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(An) is also zero, which proves the counterexample.
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.
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