From Wikipedia, the free encyclopedia
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,}
and
m
(
x
)
>
0
{\displaystyle m(x)>0}
if
x
≠
0
{\displaystyle x\neq 0}
.
If
x
≤
y
{\displaystyle x\leq y}
, then
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}}
is a
decreasing sequence with greatest lower bound 0, then the sequence
m
(
x
n
)
{\displaystyle m(x_{n})}
has
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 , Second Series, 48 (1): 154–167,
doi :
10.2307/1969222 ,
JSTOR
1969222 ,
MR
0018718 ,
Zbl
0029.20401
Talagrand, Michel (2008), "Maharam's Problem",
Annals of Mathematics , Second Series, 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