In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947 ).
Definitions
A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function m such that
m
(
0
)
=
0
,
m
(
1
)
=
1
,
{\displaystyle m(0)=0,m(1)=1,}
m
(
x
)
>
0
{\displaystyle m(x)>0}
x
≠ ≠ -->
0
{\displaystyle x\neq 0}
If
x
≤ ≤ -->
y
{\displaystyle x\leq y}
m
(
x
)
≤ ≤ -->
m
(
y
)
{\displaystyle m(x)\leq m(y)}
m
(
x
∨ ∨ -->
y
)
≤ ≤ -->
m
(
x
)
+
m
(
y
)
− − -->
m
(
x
∧ ∧ -->
y
)
{\displaystyle m(x\vee y)\leq m(x)+m(y)-m(x\wedge y)}
If
x
n
{\displaystyle x_{n}}
decreasing sequence with greatest lower bound 0, then the sequence
m
(
x
n
)
{\displaystyle m(x_{n})}
limit 0. A Maharam algebra is a complete Boolean algebra with a continuous submeasure.
Examples
Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo measure zero sets is complete, it is a Maharam algebra.
Michel Talagrand (2008 ) solved a long-standing problem by constructing a Maharam algebra that is not a measure algebra , i.e. , that does not admit any countably additive strictly positive finite measure.
References
Balcar, Bohuslav ; Jech, Thomas (2006), "Weak distributivity, a problem of von Neumann and the mystery of measurability" , Bulletin of Symbolic Logic 12 (2): 241– 266, doi :10.2178/bsl/1146620061 , MR 2223923 , Zbl 1120.03028 Maharam, Dorothy (1947), "An algebraic characterization of measure algebras", Annals of Mathematics 48 (1): 154– 167, doi :10.2307/1969222 , JSTOR 1969222 , MR 0018718 , Zbl 0029.20401 Talagrand, Michel (2008), "Maharam's Problem", Annals of Mathematics 168 (3): 981– 1009, doi :10.4007/annals.2008.168.981 JSTOR 40345433 , MR 2456888 , Zbl 1185.28002 Velickovic, Boban (2005), "CCC forcing and splitting reals", Israel Journal of Mathematics 147 : 209– 220, doi :10.1007/BF02785365 , MR 2166361 , Zbl 1118.03046
