Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."
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Measure Thoery: Chp 1.
Measures
Algebras and Sigma-Algebras
Algebras Let be an arbitrary set. A collection of subsets of is an algebra
on if
,
for each set that belongs to
, the set belongs to ,
for each finite sequence of sets that belong to , the set belongs to , and
for each finite sequence of sets that belongs to , the set belongs to .
Sigma-Algebras Let be an arbitrary set. A collection of subsets of is a -algebra on if
,
for each set that belongs to
, the set belongs to ,
for each finite sequence of sets that belong to , the set belongs to
, and
for each finite sequence of sets that belongs to , the set belongs to
.
Measures
Let be a set, and let be a -algebra on . A function whose domain is the -algebra and whose values belong to
the extended half-line
is said to be countably additive if it satisfies
for each infinite sequence of disjoint sets that belong to . A measure
(or a countably additive measure) on is a function that satisfies and is countably
additive. A triplet is often called a measure space.
Outer Measures
Let be a set, and let be the collection of all
subsets of . An outer
measure on is a function
such that
,
if ,
then , and
if is an
infinite sequence of subsets of ,
then .
Thus an outer measure on is a
monotone and countably subadditive function from to whose value at is .
-measurable (or
measurable with respect to ) Let be a set, and let be an outer measure on . A subset of is -measurable if
holds for every subset of
.
Theorem 1.3.6 Let be a set, let be an outer measure on , and let be the collection
of all -measurable subsets
of . Then - is a -algebra, and - the restriction of
to is a measure on
.
Lebesgue Measure
The Borel -algebra on is the -algebra on generated by the collection
of open subsets of ; it
is denoted by .
The Borel subsets of
are those that belong to .
Completeness and Regularity
Let be a
measure space. The measure is
complete if the relations , and together imply that . It's sometimes convenient to call a subset
of -negligible (or -null) if there is a subset of such that , and . Thus the measure is complete if and only if every
-negligible subset of belongs to .
If is an outer measure
on the set and if is the -algebra of all -measurable subsets of , then the restriction of to is
complete.
Let be a -algebra on that includes the -algebra of Borel sets.
A measure on is
regular if - each compact subset of satisfies - each set in satisfies
each open subset of satisfies
Dynkin Classes
Let be a set. A collection
of subsets of is a d-system (or a
Dynkin class) on if - , - whenever and , and - whenever is an increasing
sequence of sets in .
A collection of subsets of is
a -system on
if it is closed under the
formation of finite intersections.
References
[1]@book{cohn2013measure, title={Measure theory},
author={Cohn, Donald L}, year={2013}, publisher={Springer} }