In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, an inner measure is a
function 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 extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
, 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 ...
defined on all
subset
In mathematics, a 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 a ...
s of a set
that satisfies the following conditions:
* Null empty set: The
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
has zero inner measure (''see also:
measure zero
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...
''); 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 a_1, a_2, \ldots is called superadditive if it satisfies the inequality
a_ \geq a_n + a_m
for all m and n.
The ...
: 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 cal ...
of sets such that
for each
and
* If the measure is not finite, that is, if there exist sets
with
, then this infinity must be approached. More precisely, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
over a set
and
be a
measure 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 me ...
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 ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
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)