Delta-Ring
Given a set 👁 X
,
let 👁 F
be a nonempty
set of subsets of 👁 X
.
Then 👁 F
is a ring if,
for every pair of sets in 👁 F
,
the intersection, union, and set difference is also in 👁 F
. 👁 F
is called a 👁 delta
-ring
if 👁 F
is a ring and, for any countable collection
of sets 👁 A_n in F
,
the intersection 👁 intersection A_n
is also in 👁 F
.
A 👁 delta
-ring 👁 F
is 👁 sigma
-finite
if 👁 X
is the union of a countable collection
of sets in 👁 F
.
Given a collection 👁 S
of subsets of 👁 X
,
the 👁 delta
-ring generated by 👁 S
can be defined as the intersection of all 👁 delta
-rings containing 👁 S
. For example, the collection of bounded real Borel
sets is a 👁 delta
-ring.
More generally, if 👁 X
is a Hausdorff topological space, then the collection
of Borel sets with compact closure is a 👁 delta
-ring.
Unbounded (complex) measures are defined on 👁 delta
-rings. If 👁 F
is a 👁 sigma
-algebra, it is a 👁 delta
-ring, and if it is a 👁 delta
-ring, it is a ring.
A ring of sets in 👁 X
is also a ring in algebraic sense, if addition is defined as
and multiplication as
See also
Ring, Sigma-AlgebraThis entry contributed by Allan Cortzen
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Cite this as:
Cortzen, Allan. "Delta-Ring." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Delta-Ring.html