In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular in
measure theory
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 simil ...
, an inner measure is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
on the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of a given
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, with values in the
extended real numbers
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.
Definition
An inner measure is a
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ...
defined on all
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of a set
that satisfies the following conditions:
* Null empty set: The
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
has zero inner measure (''see also:
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
''); that is,
*
Superadditive In mathematics, a function f is superadditive if
f(x+y) \geq f(x) + f(y)
for all x and y in the domain of f.
Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality
a_ \geq a_n + a_m
for all m and n.
The ter ...
: For any
disjoint sets
and
* Limits of decreasing towers: For any
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 calle ...
of sets such that
for each
and
* Infinity must be approached: If
for a set
then for every positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
there exists some
such that
The inner measure induced by a measure
Let
be a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
over a set
and
be a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
on
Then the inner measure
induced by
is defined by
Essentially
gives a lower bound of the size of any set by ensuring it is at least as big as the
-measure of any of its
-measurable subsets. Even though the set function
is usually not a measure,
shares the following properties with measures:
#
#
is non-negative,
# If
then
Measure completion
Induced inner measures are often used in combination with
outer measure
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...
s to extend a measure to a larger σ-algebra. If
is a finite measure defined on a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
over
and
and
are corresponding induced outer and inner measures, then the sets
such that
form a σ-algebra
with
.
[Halmos 1950, § 14, Theorem F]
The set function
defined by
for all
is a measure on
known as the completion of
See also
*
References
* Halmos, Paul R., ''Measure Theory'', D. Van Nostrand Company, Inc., 1950, pp. 58.
* A. N. Kolmogorov & S. V. Fomin, translated by Richard A. Silverman, ''Introductory Real Analysis'', Dover Publications, New York, 1970, (Chapter 7)
{{Measure theory
Measures (measure theory)