A set function generally aims to measure subsets in some way.
Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Additionally, a semiring is a
π-system where every complement is equal to a finite
disjoint union of sets in
A semialgebra is a semiring where every complement is equal to a finite
disjoint union of sets in are arbitrary elements of and it is assumed that
In general, it is typically assumed that is always
well-defined for all or equivalently, that does not take on both and as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever is
finitely additive:
Set difference formula: is defined with satisfying and
Null sets
A set is called a null set (with respect to ) or simply null if
Whenever is not identically equal to either or then it is typically also assumed that:
A set function is called finite if for every the value is finite (which by definition means that and ; an infinite value is one that is equal to or ).
Every finite set function must have a finite
mass.
The series on the left hand side is defined in the usual way as the limit
As a consequence, if is any
permutation/
bijection then this is because and applying this condition (a) twice guarantees that both and hold. By definition, a convergent series with this property is said to be
unconditionally convergent. Stated in
plain English, this means that rearranging/relabeling the sets to the new order does not affect the sum of their measures. This is desirable since just as the union does not depend on the order of these sets, the same should be true of the sums and
if is not infinite then this series must also
converge absolutely, which by definition means that must be finite. This is automatically true if is
non-negative (or even just valued in the extended real numbers).
As with any convergent series of real numbers, by the
Riemann series theorem, the series converges absolutely if and only if its sum does not depend on the order of its terms (a property known as
unconditional convergence). Since unconditional convergence is guaranteed by (a) above, this condition is automatically true if is valued in
if is infinite then it is also required that the value of at least one of the series be finite (so that the sum of their values is well-defined). This is automatically true if is
non-negative.
a
measure if it is a
pre-measure whose domain is a
σ-algebra. That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a
null empty set.
complete if every subset of every
null set is null; explicitly, this means: whenever and is any subset of then and
Unlike many other properties, completeness places requirements on the set (and not just on 's values).
𝜎-finite if there exists a sequence in such that is finite for every index and also
decomposable if there exists a subfamily of pairwise disjoint sets such that is finite for every and also (where ).
Every 𝜎-finite set function is decomposable although not conversely. For example, the
counting measure on (whose domain is ) is decomposable but not 𝜎-finite.
If is valued in a
normed space then it is countably additive if and only if for any
pairwise disjoint sequence in If is finitely additive and valued in a
Banach space then it is countably additive if and only if for any pairwise disjoint sequence in
As described
in this article's section on generalized series, for any family of
real numbers indexed by an arbitrary
indexing set it is possible to define their sum as the limit of the
net of finite partial sums where the domain is
directed by
Whenever this
net converges then its limit is denoted by the symbols while if this net instead diverges to then this may be indicated by writing
Any sum over the empty set is defined to be zero; that is, if then by definition.
For example, if for every then
And it can be shown that
If then the generalized series converges in if and only if converges unconditionally (or equivalently,
converges absolutely) in the usual sense.
If a generalized series converges in then both and also converge to elements of and the set is necessarily
countable (that is, either finite or
countably infinite);
this remains true if is replaced with any
normed space.[proof 1]
It follows that in order for a generalized series to converge in or it is necessary that all but at most countably many will be equal to which means that is a sum of at most countably many non-zero terms.
Said differently, if is uncountable then the generalized series does not converge.
In summary, due to the nature of the real numbers and its topology, every generalized series of real numbers (indexed by an arbitrary set) that converges can be reduced to an ordinary absolutely convergent series of countably many real numbers. So in the context of measure theory, there is little benefit gained by considering uncountably many sets and generalized series. In particular, this is why the definition of "
countably additive" is rarely extended from countably many sets in (and the usual countable series ) to arbitrarily many sets (and the generalized series ).
Inner measures, outer measures, and other properties
continuous from above if for all non-increasing sequences of sets in such that with and all finite.
Lebesgue measure is continuous from above but it would not be if the assumption that all are eventually finite was omitted from the definition, as this example shows: For every integer let be the open interval so that where
continuous from below if for all non-decreasing sequences of sets in such that
infinity is approached from below if whenever satisfies then for every real there exists some such that and
If is a
topology on then a set function is said to be:
a
Borel measure if it is a measure defined on the σ-algebra of all
Borel sets, which is the smallest σ-algebra containing all open subsets (that is, containing ).
and are
singular, written if there exist disjoint sets and in the domains of and such that for all in the domain of and for all in the domain of
Examples
Examples of set functions include:
The function assigning
densities to sufficiently
well-behaved subsets is a set function.
A
probability measure assigns a probability to each set in a
σ-algebra. Specifically, the probability of the
empty set is zero and the probability of the
sample space is with other sets given probabilities between and
A possibility measure assigns a number between zero and one to each set in the
powerset of some given set. See
possibility theory.
The
Jordan measure on is a set function defined on the set of all Jordan measurable subsets of it sends a Jordan measurable set to its Jordan measure.
Lebesgue measure
The
Lebesgue measure on is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue -algebra.[5]
Its definition begins with the set of all intervals of real numbers, which is a
semialgebra on
The function that assigns to every interval its is a finitely additive set function (explicitly, if has endpoints then ).
This set function can be extended to the
Lebesgue outer measure on which is the translation-invariant set function that sends a subset to the
infimum
Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the
𝜎-algebra of all subsets that satisfy the
Carathéodory criterion:
is a measure that called
Lebesgue measure.
Vitali sets are examples of
non-measurable sets of real numbers.
Finitely additive translation-invariant set functions
The only translation-invariant measure on with domain that is finite on every compact subset of is the trivial set function that is identically equal to (that is, it sends every to )[6]
However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover, some are even valued in In fact, such non-trivial set functions will exist even if is replaced by any other
abeliangroup[7]
Theorem[8] — If is any
abelian group then there exists a finitely additive and translation-invariant[note 1] set function of mass
Suppose that is a set function on a
semialgebra over and let
which is the
algebra on generated by
The
archetypal example of a semialgebra that is not also an
algebra is the family
on where for all [9] Importantly, the two non-strict inequalities in cannot be replaced with strict inequalities since semialgebras must contain the whole underlying set that is, is a requirement of semialgebras (as is ).
If is
finitely additive then it has a unique extension to a set function on defined by sending (where indicates that these are
pairwise disjoint) to:[9]
This extension will also be finitely additive: for any pairwise disjoint [9]
If in addition is extended real-valued and
monotone (which, in particular, will be the case if is
non-negative) then will be monotone and
finitely subadditive: for any such that [9]
To define this extension, first extend to an
outer measure on by
and then restrict it to the set of -measurable sets (that is,
Carathéodory-measurable sets), which is the set of all such that
It is a -algebra and is sigma-additive on it, by Caratheodory lemma.
^The function being translation-invariant means that for every and every subset
Proofs
^Suppose the net converges to some point in a
metrizable topological vector space (such as or a
normed space), where recall that this net's domain is the
directed set
Like every convergent net, this convergent net of partial sums is a Cauchy net, which for this particular net means (by definition) that for every neighborhood of the origin in there exists a finite subset of such that
for all finite supersets
this implies that for every (by taking and ).
Since is metrizable, it has a countable neighborhood basis at the origin, whose intersection is necessarily (since is a Hausdorff TVS).
For every positive integer pick a finite subset such that for every
If belongs to then belongs to
Thus for every index that does not belong to the countable set
A set function generally aims to measure subsets in some way.
Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Additionally, a semiring is a
π-system where every complement is equal to a finite
disjoint union of sets in
A semialgebra is a semiring where every complement is equal to a finite
disjoint union of sets in are arbitrary elements of and it is assumed that
In general, it is typically assumed that is always
well-defined for all or equivalently, that does not take on both and as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever is
finitely additive:
Set difference formula: is defined with satisfying and
Null sets
A set is called a null set (with respect to ) or simply null if
Whenever is not identically equal to either or then it is typically also assumed that:
A set function is called finite if for every the value is finite (which by definition means that and ; an infinite value is one that is equal to or ).
Every finite set function must have a finite
mass.
The series on the left hand side is defined in the usual way as the limit
As a consequence, if is any
permutation/
bijection then this is because and applying this condition (a) twice guarantees that both and hold. By definition, a convergent series with this property is said to be
unconditionally convergent. Stated in
plain English, this means that rearranging/relabeling the sets to the new order does not affect the sum of their measures. This is desirable since just as the union does not depend on the order of these sets, the same should be true of the sums and
if is not infinite then this series must also
converge absolutely, which by definition means that must be finite. This is automatically true if is
non-negative (or even just valued in the extended real numbers).
As with any convergent series of real numbers, by the
Riemann series theorem, the series converges absolutely if and only if its sum does not depend on the order of its terms (a property known as
unconditional convergence). Since unconditional convergence is guaranteed by (a) above, this condition is automatically true if is valued in
if is infinite then it is also required that the value of at least one of the series be finite (so that the sum of their values is well-defined). This is automatically true if is
non-negative.
a
measure if it is a
pre-measure whose domain is a
σ-algebra. That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a
null empty set.
complete if every subset of every
null set is null; explicitly, this means: whenever and is any subset of then and
Unlike many other properties, completeness places requirements on the set (and not just on 's values).
𝜎-finite if there exists a sequence in such that is finite for every index and also
decomposable if there exists a subfamily of pairwise disjoint sets such that is finite for every and also (where ).
Every 𝜎-finite set function is decomposable although not conversely. For example, the
counting measure on (whose domain is ) is decomposable but not 𝜎-finite.
If is valued in a
normed space then it is countably additive if and only if for any
pairwise disjoint sequence in If is finitely additive and valued in a
Banach space then it is countably additive if and only if for any pairwise disjoint sequence in
As described
in this article's section on generalized series, for any family of
real numbers indexed by an arbitrary
indexing set it is possible to define their sum as the limit of the
net of finite partial sums where the domain is
directed by
Whenever this
net converges then its limit is denoted by the symbols while if this net instead diverges to then this may be indicated by writing
Any sum over the empty set is defined to be zero; that is, if then by definition.
For example, if for every then
And it can be shown that
If then the generalized series converges in if and only if converges unconditionally (or equivalently,
converges absolutely) in the usual sense.
If a generalized series converges in then both and also converge to elements of and the set is necessarily
countable (that is, either finite or
countably infinite);
this remains true if is replaced with any
normed space.[proof 1]
It follows that in order for a generalized series to converge in or it is necessary that all but at most countably many will be equal to which means that is a sum of at most countably many non-zero terms.
Said differently, if is uncountable then the generalized series does not converge.
In summary, due to the nature of the real numbers and its topology, every generalized series of real numbers (indexed by an arbitrary set) that converges can be reduced to an ordinary absolutely convergent series of countably many real numbers. So in the context of measure theory, there is little benefit gained by considering uncountably many sets and generalized series. In particular, this is why the definition of "
countably additive" is rarely extended from countably many sets in (and the usual countable series ) to arbitrarily many sets (and the generalized series ).
Inner measures, outer measures, and other properties
continuous from above if for all non-increasing sequences of sets in such that with and all finite.
Lebesgue measure is continuous from above but it would not be if the assumption that all are eventually finite was omitted from the definition, as this example shows: For every integer let be the open interval so that where
continuous from below if for all non-decreasing sequences of sets in such that
infinity is approached from below if whenever satisfies then for every real there exists some such that and
If is a
topology on then a set function is said to be:
a
Borel measure if it is a measure defined on the σ-algebra of all
Borel sets, which is the smallest σ-algebra containing all open subsets (that is, containing ).
and are
singular, written if there exist disjoint sets and in the domains of and such that for all in the domain of and for all in the domain of
Examples
Examples of set functions include:
The function assigning
densities to sufficiently
well-behaved subsets is a set function.
A
probability measure assigns a probability to each set in a
σ-algebra. Specifically, the probability of the
empty set is zero and the probability of the
sample space is with other sets given probabilities between and
A possibility measure assigns a number between zero and one to each set in the
powerset of some given set. See
possibility theory.
The
Jordan measure on is a set function defined on the set of all Jordan measurable subsets of it sends a Jordan measurable set to its Jordan measure.
Lebesgue measure
The
Lebesgue measure on is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue -algebra.[5]
Its definition begins with the set of all intervals of real numbers, which is a
semialgebra on
The function that assigns to every interval its is a finitely additive set function (explicitly, if has endpoints then ).
This set function can be extended to the
Lebesgue outer measure on which is the translation-invariant set function that sends a subset to the
infimum
Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the
𝜎-algebra of all subsets that satisfy the
Carathéodory criterion:
is a measure that called
Lebesgue measure.
Vitali sets are examples of
non-measurable sets of real numbers.
Finitely additive translation-invariant set functions
The only translation-invariant measure on with domain that is finite on every compact subset of is the trivial set function that is identically equal to (that is, it sends every to )[6]
However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover, some are even valued in In fact, such non-trivial set functions will exist even if is replaced by any other
abeliangroup[7]
Theorem[8] — If is any
abelian group then there exists a finitely additive and translation-invariant[note 1] set function of mass
Suppose that is a set function on a
semialgebra over and let
which is the
algebra on generated by
The
archetypal example of a semialgebra that is not also an
algebra is the family
on where for all [9] Importantly, the two non-strict inequalities in cannot be replaced with strict inequalities since semialgebras must contain the whole underlying set that is, is a requirement of semialgebras (as is ).
If is
finitely additive then it has a unique extension to a set function on defined by sending (where indicates that these are
pairwise disjoint) to:[9]
This extension will also be finitely additive: for any pairwise disjoint [9]
If in addition is extended real-valued and
monotone (which, in particular, will be the case if is
non-negative) then will be monotone and
finitely subadditive: for any such that [9]
To define this extension, first extend to an
outer measure on by
and then restrict it to the set of -measurable sets (that is,
Carathéodory-measurable sets), which is the set of all such that
It is a -algebra and is sigma-additive on it, by Caratheodory lemma.
^The function being translation-invariant means that for every and every subset
Proofs
^Suppose the net converges to some point in a
metrizable topological vector space (such as or a
normed space), where recall that this net's domain is the
directed set
Like every convergent net, this convergent net of partial sums is a Cauchy net, which for this particular net means (by definition) that for every neighborhood of the origin in there exists a finite subset of such that
for all finite supersets
this implies that for every (by taking and ).
Since is metrizable, it has a countable neighborhood basis at the origin, whose intersection is necessarily (since is a Hausdorff TVS).
For every positive integer pick a finite subset such that for every
If belongs to then belongs to
Thus for every index that does not belong to the countable set