In mathematics, a Maharam algebra is a

complete Boolean algebra In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boole ...

with a continuous submeasure (defined below). They were introduced by .

Definitions

A continuous submeasure or Maharam submeasure on a

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

is a

real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...

''m'' such that *

m(0)=0, m(1)=1,

and

m(x)>0

x\ne 0

. * If

x\le y

, then

m(x)\le m(y)

. *

m(x\vee y)\le m(x)+m(y)-m(x\wedge y)

. * If

x_n

is a

decreasing sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called th ...

with greatest lower bound 0, then the sequence

m(x_n)

has

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

0. A Maharam algebra is a

with a continuous submeasure.

Examples

Every

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...

is a continuous submeasure, so as the corresponding Boolean algebra of

measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...

s modulo measure zero sets is complete, it is a Maharam algebra. solved a long-standing problem by constructing a Maharam algebra that is not a

measure algebra In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets. Definition A measure alg ...

, ''i.e.'', that does not admit any countably additive strictly positive finite measure.

References

* * * * Boolean algebra {{algebra-stub