measure on a Boolean algebra

Let A be a Boolean algebraMathworldPlanetmath. A measureMathworldPlanetmath on A is a non-negative extended real-valued function m defined on A such that

  1. 1.

    there is an aA such that m(a) is a real number (not ),

  2. 2.

    if ab=0, then m(ab)=m(a)+m(b).

For example, a sigma algebra over a set E is a Boolean algebra, and a measure ( μ on the measurable spaceMathworldPlanetmathPlanetmath (,E) is a measure on the Boolean algebra .

The following are some of the elementary properties of m:

  • m(0)=0.

    By condition 1, suppose m(a)=r, then m(a)=m(0a)=m(0)+m(a), so that m(0)=0.

  • m is non-decreasing: m(a)m(b) for ab

    If ab, then c=b-a and a are disjoint (ca=0) and b=ca. So m(b)=m(ca)=m(c)+m(a). As a result, m(a)m(b).

  • m is subadditive: m(ab)m(a)+m(b).

    Since ab=(a-b)b, and a-b and b are disjoint, we have that m(ab)=m((a-b)b)=m(a-b)+m(b). Since a-ba, the result follows.

From the three properties above, one readily deduces that I:={aAm(a)=0} is a Boolean ideal of A.

A measure on A is called a two-valued measure if m maps onto the two-element set {0,1}. Because of the existence of an element aA with m(a)=1, it follows that m(1)=1. Consequently, the set F:={aAm(a)=1} is a Boolean filter. In fact, because m is two-valued, F is an ultrafilterMathworldPlanetmath (and correspondingly, the set {am(a)=0} is a maximal ideal).

Conversely, given an ultrafilter F of A, the function m:A{0,1}, defined by m(a)=1 iff aF, is a two-valued measure on A. To see this, suppose ab=0. Then at least one of them, say a, can not be in F (or else 0=abF). This means that m(a)=0. If bF, then abF, so that m(ab)=1=m(b)=m(b)+m(a). On the other hand, if bF, then a,bF, so abF, or abF. This means that m(ab)=0=m(a)+m(b).

Remark. A measure (on a Boolean algebra) is sometimes called finitely additive to emphasize the defining condition 2 above. In additionPlanetmathPlanetmath, this terminology is used when there is a need to contrast a stronger form of additivity: countable additivity. A measure is said to be countably additive if whenever K is a countable set of pairwise disjoint elements in A such that K exists, then

Title measure on a Boolean algebra
Canonical name MeasureOnABooleanAlgebra
Date of creation 2013-03-22 17:59:16
Last modified on 2013-03-22 17:59:16
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06B99
Related topic Measure
Defines measure
Defines two-valued measure
Defines finitely additive
Defines countably additive