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ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS

S. CAMBRONERO 1, D. CAMPOS 2, C. A. FONSECA-MORA 3, AND D. MENA 4

Abstract. The concept of supremum of a family of set functions was introduced by M.

Veraar and I. Yaroslavtsev (2016) for families of measures defined on a measurable space.

We expand this concept to include families of set functions in a very general setting. The

case of families of signed measures is widely discussed and exploited.

Contents

1. Introduction 1

2. The set function supremum 2

3. Sub-additive set functions on rings 6

4. Supremum of signed measures 8

4.1. Signed measures with density 11

4.2. Application: Vector measures on a space of bilinear forms 13

Declarations 14

References 14

1. Introduction

Given a family of sets A and a family {fj}j∈J of real extended set functions defined on A,

the classical supremum defined as f(A) := sup{fj(A) : A ∈ A} gives the least upper bound

of the family under the natural order (see Definition 2.8).

In practical situations, one might need to work with a smaller class of set functions, as

for instance the class of measures over a sigma-algebra, or the class of finitely additive set

functions over an algebra. The classical definition of supremum given above might produce

an object which is not in the class.

2020 Mathematics Subject Classification. 28A10, 28B05.

Key words and phrases. set function supremum; admissibility; signed measures.
1

http://arxiv.org/abs/2308.10914v1
https://orcid.org/0000-0001-6758-4942
https://orcid.org/0000-0002-3608-1151
https://orcid.org/0000-0002-9280-8212
https://orcid.org/0000-0002-9443-391X


ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 2

In [9], they show that, in the class of (non-negative) measures on a measurable space (Ω,F),

a suitable definition for the supremum of a family of measures (νj)j∈J is

µ(A) = sup
Π∈P(A)

∑

C∈Π

sup
j∈J

νj(C),

where P(A) is the family of all finite partitions of A by elements of F .

In this paper, we show that this definition works perfectly well in larger classes of set

functions and explore some nice consequences of this concept.

Although the literature provides plenty of concepts and results related to that of the

supremum of measures, we could only find the first reference in [9]. Some works like [5] and

[7] have mentioned the existence of a supremum in the case of a family of measures, but not

in a constructive way. In [9], they do provide a constructive definition for this particular

kind of families, developing a few tools they needed in their construction of the quadratic

variation of a cylindrical local martingale. When trying to apply this concept to our recent

work on cylindrical martingale-valued measures [1], we have found it necessary to extend

it to families of a more general class of set functions, for instance to families of random

measures, signed measures or even random sub-additive set functions.

In Section 2 we give the definition of the supremum of a family of set functions in the most

general possible setting. We establish three levels of admissibility, that will be exploited

according to the kind of properties we need the supremum to inherit from the given family.

The weak admissibility is the least we can ask in order to define the set function supremum,

but we need admissibility if we want this function to inherit at least finitely sub-additivity

from the given family.

In Section 3 we work with set functions defined on rings. The main result on this section,

Theorem 3.4, shows that the set function supremum of an admissible family of sub-additive

set functions is a signed measure, whenever we can control it from below. We also introduce

the concept of set function infimum.

Results in Section 4 can be considered as the main contributions of the paper. We

concentrate on families of signed measures, for which the concepts of admissibility and

strong admissibility are equivalent. We use the set function supremum to express the

positive and negative part of signed measure, as well as the total variation. The main result,

Theorem 4.11, generalizes a result stated on [9] for families of (non-negative) measures. We

finish by giving some very pleasant applications of this result.

2. The set function supremum

Throughout this section we consider a non-empty set S and a family A of subsets of S.

Unless otherwise stated, no particular structure is assumed on A. We denote by C the class

of all set functions f : A → R.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 3

Definition 2.1. Consider a family (fj)j∈J on C. We say that this family is:

(i) weakly admissible if, for any pair of disjoint sets C,D ∈ A,

sup
j∈J

fj(D) = ∞ ⇒ sup
j∈J

fj(C) > −∞.

(ii) admissible if for each C ∈ A we have

sup
j∈J

fj(C) > −∞.

(iii) strongly admissible if there is a ∈ R such that

∀C ∈ A, sup
j∈J

fj(C) > a.

It is evident that every strongly admissible family is admissible, and every admissible family

is weakly admissible. The importance of these properties will become clear as we establish

and discuss the definition of set function supremum. Before that, we have the following

example.

Example 2.2. For A ∈ B(R) let |A| denote its Lebesgue measure.

(i) For every n ∈ N let αn : B(R) → R be defined by

αn(A) := |A ∩ (0, n]| − |A ∩ (−n, 0]| .

The family (αn)n≥1 is strongly admissible.

(ii) Let g : B(R) → R be defined by

g(A) = |A ∩ (0,∞)| − |A ∩ (−∞, 0)| , ∀A ∈ B(R), A 6= R,

and g(R) = 0. The single element family (g) is not weakly admissible since for

C = (−∞, 0) and D = (0,∞) we have g(D) = ∞ but g(C) = −∞.

The following results state useful sufficient conditions for admissibility and for strong

admissibility. Their proofs are simple so we omit them.

Lemma 2.3. Any family (fj)j∈J of functions from A into (−∞,∞] is admissible.

Lemma 2.4. Consider a family (fj)j∈J on C. Assume that there exists a ∈ R and j0 ∈ J

such that fj0(A) ≥ a for every A ∈ A. Then the family (fj)j∈J is strongly admissible. In

particular, any family on C for which at least one of its members is non-negative, is strongly

admissible.

Definition 2.5. Consider a weakly admissible family (fj)j∈J on C. We define its set function

supremum as
(

ssup
j∈J

fj

)

(A) := sup
Π∈P(A)

∑

C∈Π

sup
j∈J

fj(C), ∀A ∈ A,



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 4

where P(A) is the family of all finite partitions of A by elements of A.

Remark 2.6. The weak admissibility condition avoids having undetermined sums in the last

definition. When the family is admissible, the supremum does not achieve the value −∞.

Example 2.7. Consider S = {0, 1} and A = 2S. For n = 1, 2, . . . define αn = n · δ{0} −

∞ · δ{1} (here ∞ · 0 = 0). Each αn is clearly a signed measure on (S, 2S). But notice that

supn≥1 αn({0}) = +∞ and supn≥1 αn({1}) = −∞. This family is not weakly admissible so

we are not able to apply the previous definition for A = S.

It is clear from the definition that ssupj∈J fj ∈ C. In order to explore more properties of the

set function supremum we will need to be able to compare elements in C.

Definition 2.8. Given f, g ∈ C, we write f ≤ g whenever f(A) ≤ g(A), ∀A ∈ A. If this is

the case, we say that g dominates f.

It is clear that the relation “≤” introduces a partial order on C. We will show (see Corollary

2.13 below) that the set function supremum of an admissible family is the smallest finitely

super-additive set function that dominates each fj. Here, a function g ∈ C is called finitely

super-additive if whenever A,B ∈ A are disjoint and A∪B ∈ A, then g(A) + g(B) is a well

defined extended real number and g(A) + g(B) ≤ g(A ∪ B).

Lemma 2.9. Let (fj)j∈J be weakly admissible and f = ssupj∈J fj. If g ∈ C is finitely

super-additive and dominates each fj, then it also dominates f .

Proof. Let A ∈ A. For any Π ∈ P(A) we have
∑

C∈Π

sup
j∈J

fj(C) ≤
∑

C∈Π

g(C) ≤ g(A),

the last inequality is a consequence of the finite super-additivity of g. Since f(A) is the

supremum of the left-hand side, this implies f(A) ≤ g(A). �

In order to talk about super-additivity of the set function supremum, weakly admissibility

of the family is not enough.

Example 2.10. Consider S = {1, 2, 3, ...} and A = 2S. Define f on A by f(∅) = 0,

f(A) = 1 if 1 /∈ A 6= ∅, f(A) = −∞ if 1 ∈ A. Let A = {1} and B = {2, 3, . . .}. The unitary

family {f} is weakly admissible but not admissible. The supremum f̌ = ssup{f}, though

well defined, is not supper-additive. In fact, notice that f̌(A) = −∞ and f̌(B) = +∞.

Lemma 2.11. If (fj)j∈J is admissible, then f = ssupj∈J fj is finitely super-additive and

dominates each fj.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 5

Proof. Let A,B ∈ A be disjoint and such that A ∪ B ∈ A. Because of the admissibility,

f(A), f(B) ∈ (−∞,∞]. Let c ∈ R such that c < f(A) + f(B). Let a < f(A) and b < f(B)

such that a + b = c. Choose partitions ΠA ∈ P(A) and ΠB ∈ P(B) such that

a <
∑

C∈ΠA

sup
j∈J

fj(C), b <
∑

C∈ΠB

sup
j∈J

fj(C).

Then for each C ∈ ΠA ∪ ΠB, there exists jC ∈ J such that

a <
∑

C∈ΠA

fjC (C), b <
∑

C∈ΠB

fjC(C).

Since ΠA ∪ ΠB ∈ P(A ∪ B), it follows that

c = a+ b <
∑

C∈ΠA∪ΠB

fjC (C) ≤ f(A ∪ B).

The inequality

f(A) + f(B) ≤ f(A ∪ B)

follows from the fact that c < f(A) + f(B) was arbitrary. �

The results of these lemmas can be applied to quite general situations. For instance, A

could be so small that no partition other that Π = {A} exists for each set A.

Example 2.12. Consider S = {0, 1, 2, 3} and A = {{0}, {0, 1}, S}. Any set function f

defined on A is additive. In fact, there are no disjoint sets on A to test. In this case,
(

ssupj∈J fj
)

(A) = supj∈J fj(A). We get a similar situation with A = {{0}, {1, 2}, S}. In

this case we have two disjoint sets, but their union is not in A.

By combining the results of Lemmas 2.9 and 2.11 we conclude the following:

Corollary 2.13. For an admissible family (fj)j∈J , the supremum ssupj∈J fj is the smallest

finitely super-additive set function that dominates each fj.

Remark 2.14. Notice the special case of a unitary family {f}, with f > −∞. In this case,

ssup{f} is the smallest finitely super-additive set function defined on A that dominates f .

In particular, f = ssup{f} if and only if f is finitely super-additive.

Remark 2.15. Consider an admissible family of set functions (νj)j∈J , defined on A ⊆ 2S.

(i) If for each C ∈ A there exists j ∈ J such that νj(C) ≥ 0, then

ssup
j∈J

νj = ssup
j∈J

(νj)+ = ssup {νj : j ∈ J} ∪ {0},

where (νj)+(A) = max{νj(A), 0}.

(ii) Given k ∈ J define µj(A) := max {νj(A), νk(A)} and λj := ssup {νj , νk}. Then

ssup
j∈J

νj = ssup
j∈J

µj = ssup
j∈J

λj.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 6

(iii) If for each C ∈ A

sup
j∈J

νj(C) ≥ − inf
j∈J

νj(C),

then,

ssup
j∈J

νj = ssup
j∈J

|νj | .

3. Sub-additive set functions on rings

In this section we assume that the family A is a ring. More precisely, A is closed under

finite unions and difference of sets, and therefore it is also closed under finite intersections.

We will try to prove that ssup fj inherits the sub-additivity from the fj ’s. Since negative

values are allowed, in the definition of sub-additivity we must assume that the sets are

disjoint. We say that a function f ∈ C is sub-additive if

f

(

∞
⋃

n=1

Bn

)

≤
∞
∑

n=1

f(Bn) (3.1)

for any countable family (Bn) of pairwise disjoint sets on A such that ∪Bn ∈ A. As part

of the definition, the sum on the right-hand side must make sense independently of any

reordering. Finite sub-additivity is defined analogously.

Lemma 3.1. Consider an admissible family (fj)j∈J of functions defined on a ring A. If

each fj is sub-additive, then f := ssup fj is also sub-additive.

Proof. Let (Bn) be a sequence of pairwise disjoint sets on A such that B = ∪Bn ∈ A. If

Π ∈ P(B), it follows that C ∩ Bn ∈ A for each C ∈ Π. By applying the sub-additivity of

fj , and the super-additivity of f one gets

∑

C∈Π

sup
j∈J

fj(C) ≤
∑

C∈Π

sup
j∈J

∞
∑

n=1

fj(C ∩ Bn) ≤
∞
∑

n=1

∑

C∈Π

f(C ∩ Bn) ≤
∞
∑

n=1

f(Bn).

Taking the supremum on the left we get the result. �

As the next example illustrates, in Lemma 3.1 the admissibility assumption can not be

replaced by weak admissibility.

Example 3.2. Consider S = {1, 2, . . .}, A = 2S and define

αn(A) =
∑

k∈A

pnk,

where pnk = (−1)k for k ≤ n and pnk = −1 for k > n. The sum is always well defined and

gives αn(A) = −∞ for any infinite set A. Each αn is sub-additive, and better yet, a signed

measure. The family (αn) is weakly admissible but not admissible. For µ = ssupαn we have

µ({k}) = (−1)k, consequently µ is not sub-additive.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 7

Lemma 3.3. Suppose the family (fj)j∈J is admissible and each fj is finitely sub-additive on

a ring A. Then f := ssupj∈J fj is finitely additive.

Proof. The argument of Lemma 3.1 clearly works for finite families of disjoint sets. This

gives the finite sub-additivity, and Lemma 2.11 finishes the work. �

Recall that, if a set function defined on a measurable space (S,F) is sub-additive, non-

negative and finitely additive, then it is a measure. This fact is used in the proof of the

following useful result.

Theorem 3.4. Consider an admissible family of sub-additive set functions defined on a

measurable space (S,F). Let µ be the set function supremum of that family, and suppose

there is a finite measure ν such that µ ≥ −ν. Then µ is a signed measure. In particular, if

µ ≥ 0, then it is a measure.

Proof. By Lemmas 3.1 and 3.3, µ is finitely additive and sub-additive, then so is the non-

negative set function µ + ν. We conclude that µ + ν is a measure, so µ = (µ + ν) − ν is a

signed measure. �

Corollary 3.5. Let (νj)j∈J be a family of sub-additive set functions defined on a measurable

space (S,F). If for some j ∈ J we have νj ≥ 0, then ν := ssupj∈J νj is a measure. In

particular, the set function supremum of a family of measures on (S,Σ), is itself a measure.

Proof. Since the family (nj) has a non-negative element, then it is strongly admissible

(Lemma 2.4) and clearly its supremum is non-negative. Hence by Theorem 3.4 its supremum

is a measure. The second assertion is an easy consequence of the former. �

Remark 3.6. The result in Corollary 3.5 generalizes Lemma 2.6 in [9].

We finish this section by introducing the concept of set function infimum. Consider a family

(αj)j∈J of set functions defined on a family of setsA. If the family (−αj) is weakly admissible,

we can define

iinf
j∈J

αj := − ssup
j∈J

(−αj).

We immediately get
(

iinf
j∈J

αj

)

(A) = inf
Π∈P(A)

∑

C∈Π

inf
j∈J

αj(C).

If (−αj) is admissible, then iinfj∈J αj is the biggest finitely sub-additive set function that is

dominated by each αj . If A is a ring and each αj is finitely super-additive, then iinfj∈J αj

is finitely additive; if each αj is super-additive, then iinfj∈J αj is super-additive.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 8

4. Supremum of signed measures

We now consider the supremum µ := ssupj∈J αj of an admissible family of signed measures

defined on a σ-algebra A.

Example 4.1. Consider the (admissible) family of finite signed measures (αn)n≥1 defined

on (R,B(R)) by

αn(A) := |A ∩ (0, n]| − |A ∩ (−n, 0]| .

Since µ̌ = ssupn≥1 αn is finitely additive we have

µ̌(A) = µ̌(A ∩ R−) + µ̌(A ∩ R+) = |A ∩ R+| − |A ∩ (−1, 0]| .

This is clearly a signed measure.

Example 4.2. Consider now the (strongly admissible) family of finite signed measures

(αn)n≥1 given by

αn(A) := |A ∩ (n− 1, n]| − |A ∩ (−n,−n + 1]| .

In this case µ̌ = ssupn≥1 αn is the measure given by µ̌(A) = |A ∩ R+|.

Since signed measures are sub-additive and finitely additive, by Theorem 3.4 µ is a signed

measure if (and only if) there exists a finite measure ν such that µ ≥ −ν. In particular, if

some member of the family misses the value −∞, then µ is a signed measure on A. The

following lemma shows that in the present case, admissibility does the whole work.

For a signed measure α, we consider the Jordan decomposition α = α+ − α−.

Lemma 4.3. Consider a family (νj)j∈J of signed measures defined on a measurable space

(S,F). The following are equivalent:

(i) The family (νj) is admissible

(ii) The family (νj) is strongly admissible

(iii) There exists k ∈ J such that νk > −∞.

Proof. If the family is admissible, by definition there exists k ∈ J such that νk(S) > −∞.

The additivity implies that νk misses the value −∞. This proves that (i) implies (iii). To

see that (iii) implies (ii) it is enough to take

a = −ν−
k (S).

Finally, it is clear that (ii) implies (i). �

Last lemma, combined with Theorem 3.4, gives us the following result.

Theorem 4.4. If (νj)j∈J is an admissible family of signed measures defined on a measurable

space (S,F), then µ := ssup νj is a signed measure with µ− < ∞.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 9

We now use the concept of set function supremum to provide alternative expressions for α+,

α− and var(α) = α+ +α−. We start by noticing that, for a signed measure α, ssup {α, 0} is

the measure defined by

ssup {α, 0}(A) := sup
Π

∑

C∈Π

max{α(C), 0} = sup
Π

∑

C∈Π

α(C)+.

Lemma 4.5. For a signed measure α we have

α+ = ssup{α, 0}, α− = ssup {−α, 0}.

Proof. In fact, let Ω+ and Ω− be respectively the positive and negative sets of a Hahn

decomposition for α. By considering the partition Π = {A ∩ Ω+, A ∩ Ω−} we have

ssup{α, 0}(A) ≥ α(A ∩ Ω+) = α+(A).

On the other hand, since for every A ∈ A we have α+(A) ≥ 0 and also

α(A) = α+(A)− α−(A) ≤ α+(A),

we must have ssup{α, 0}(A) ≤ a+(A), and thus the desired equality. To get the second

identity, we apply what we have just proven to −α. �

Remark 4.6. If |α| denotes the absolute value of α, which is a sub-additive set function,

then var(α) = α+ + α− coincides with ssup {|α|} (the supremum of a unitary family). This

readily follows from the definition of supremum of measures and the general definition of

variation for a signed measure.

Lemma 4.7. For a signed measure α we have

var(α) = ssup{α,−α}

Proof. In fact, by definition we have

ssup{α,−α}(A) = sup
Π

∑

C∈Π

max{α(C),−α(C)} = sup
Π

∑

C∈Π

|α(C)| = var(α). �

Because of Theorem 3.4, when (−αj) is an admissible family of sub-additive set functions,

then µ := iinfj∈J αj is a signed measure if and only if there exists a finite measure ν such

that µ ≤ ν. In particular, if there exists j such that αj < ∞, then µ is a signed measure.

More particularly, if (µj)j∈J is a family of measures and (−µj) is admissible, according to

Theorem 4.4, iinfj∈J νj is also a measure.

Theorem 4.8. Consider a family (νj)j∈J of finite signed measures on the measurable space

(Ω,F). The signed measure µ := ssupj∈J νj satisfies

µ+ = ssup
j∈J

ν+
j , µ− ≤ iinf

j∈J
ν−
j .



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 10

If J is countable, we also have

µ− = iinf
j∈J

ν−
j .

Proof. It is clear that µ ≤ ssup ν+
j . Let Ω+ and Ω− form a Hahn decomposition of µ. We

have µ ≥ 0 and µ ≥ νj on Ω+ and consequently µ ≥ ssup{νj, 0} = ν+
j on Ω+. This shows

that on Ω+ we have µ+ = µ = ssup ν+
j . On Ω− we have νj ≤ µ ≤ 0 and consequently ν+

j = 0

for each j. We conclude µ = ssup ν+
j on Ω+ and on Ω−, so all over Ω.

Now, on Ω− we also have ν−
j = ssup{−νj , 0} = −νj and then

µ− = −µ = − ssup νj = iinf(−νj) = iinf ν−
j

while on Ω+

0 = µ− ≤ iinf
j∈J

ν−
j .

It remains to show that, for the countable case, iinf νj = 0 on Ω+. In that case, we can

choose a Hahn decomposition {Ω+
j ,Ω

−
j } for each νj, it is easy to check that Ω+ = ∪Ω+

j and

Ω− = ∩Ω−
j is a Hahn decomposition for µ. Since iinf ν−

j ≤ ν−
k = 0 on Ω+

k for each k ∈ J ,

the result follows. �

Using an analogous argument, we can infer the next result.

Corollary 4.9. Consider a family (νj)j∈J of finite signed measures on the measurable space

(Ω,F). The signed measure µ := iinfj∈J νj satisfies

µ− = ssup
j∈J

ν−
j , µ+ ≤ iinf

j∈J
ν+
j .

If J is countable, we also have

µ+ = iinf
j∈J

ν+
j .

Theorem 4.10. Consider a family (νj)j∈J of finite signed measures on the measurable space

(Ω,F), and let µ = ssupj∈J νj. Then

var(µ) ≤ ssup
j∈J

|νj| = ssup
j∈J

var(νj).

Proof. Since for all A ∈ F we have

µ(A) = (ssup
j∈J

νj)(A) ≤ (ssup
j∈J

|νj|)(A),

−µ(A) = (iinf
j∈J

(−νj))(A) ≤ (iinf
j∈J

|νj |)(A) ≤ (ssup
j∈J

|νj|)(A),

then |µ| ≤ ssupj∈J |νj |. It follows that var(µ) = ssup{|µ|} ≤ ssupj∈J |νj |. Now, since for

every j ∈ J we have |νj | ≤ var(νj) then ssupj∈J |νj| ≤ ssupj∈J var(νj). On the other hand,



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 11

for every j ∈ J , |νj | ≤ ssupj∈J |νj |, so var(νj) = ssup{|νj |} ≤ ssupj∈J |νj|. By definition of

supremum of measures we get

ssup
j∈J

var(νj) ≤ ssup
j∈J

|νj|

and therefore we have the desired equality. �

4.1. Signed measures with density. We extend Lemma 2.8 on [9] to the case of signed

measures.

Theorem 4.11. Let (S,A, ν) be a σ-finite measure space. Let F be a family of measurable

functions from S into (−∞,∞], such that
ˆ

f−dν < ∞ for each f ∈ F.

Let (fj)j∈N be a sequence in F . Define f̌ = supj≥1 fj and assume that supf∈F f = f̌ . For

each f ∈ F let µf be the signed measure defined by

µf(A) =

ˆ

B

fdν, A ∈ A.

If we define µ̌ := ssupf∈F µf , then µ̌ = ssupj≥1 µfj and

µ̌(A) =

ˆ

A

f̌dν. (4.1)

Proof. Clearly f̌ is measurable, so A 7→
´

A
f̌dν defines a signed measure that dominates

each µf . This gives the inequality “≤” in (4.1).

For the other inequality, we adapt the proof of Lemma 2.8 in [9], which is established for

non-negative functions and measures. Let A ∈ A, n ∈ N and 0 < ε < 1. Define

A1 = {s ∈ A : f1(s) > (f̌(s) ∧ n)(1− ε) > 0}

and

Aj+1 = {s ∈ A : fj+1(s) > (f̌(s) ∧ n)(1− ε) > 0} \

j
⋃

i=1

Ai.

The sets Aj , j ≥ 1 are disjoint and cover A ∩ {f̌ > 0}, so

µ̌(A ∩ {f̌ > 0}) =
∑

j≥1

µ̌(Aj) ≥
∑

j≥1

µfj(Aj) =
∑

j≥1

ˆ

Aj

fjdν.

It follows that

µ̌(A ∩ {f̌ > 0}) ≥ (1− ε)
∑

j≥1

ˆ

Aj

(f̌ ∧ n)dν = (1− ε)

ˆ

A∩{f̌>0}

(f̌ ∧ n)dν.



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 12

Since ε and n are arbitrary, we obtain

µ̌(A ∩ {f̌ > 0}) ≥

ˆ

A∩{f̌>0}

f̌dν.

If we now use

A1 = {s ∈ A : f̌(s) < 0, f1(s) > f̌(s)(1 + ε)}

and

Aj+1 = {s ∈ A : f̌(s) < 0, fj+1(s) > f̌(s)(1 + ε)} \

j
⋃

i=1

Ai

we get

µ̌(A ∩ {f̌ < 0}) ≥

ˆ

A∩{f̌<0}

f̌dν.

Now with A1 = {s ∈ A : −ε < f1(s) ≤ 0} and

Aj+1 = {s ∈ A : −ε < fj+1(s) ≤ 0} \

j
⋃

i=1

Ai

we get

µ̌(A ∩ {f̌ = 0}) ≥ −εν(A ∩ {f̌ = 0}).

Since ε is arbitrary and (by localization) ν may be assumed finite, we get µ̌(A∩{f̌ = 0}) ≥ 0.

Finally

µ̌(A) = µ̌(A ∩ {f̌ > 0}) + µ̌(A ∩ {f̌ = 0}) + µ̌(A ∩ {f̌ < 0}) ≥

ˆ

A

f̌dν.

We have proven (4.1). If we replace F with (fj), we obtain

ssup
j≥1

µfj =

ˆ

A

f̌dν = µ̌

and this completes the proof. �

Corollary 4.12. Consider a signed measure µ given by

µ(A) :=

ˆ

A

f(s) dν(s),

where ν is a σ-finite measure. Then

µ+(A) =

ˆ

A

f+(s) dν(s), µ−(A) =

ˆ

A

f−(s) dν(s), var(µ)(A) =

ˆ

A

|f(s)| dν(s).

Proof. By Theorem 4.11 we have

µ+(A) = ssup{µ, 0}(A) =

ˆ

A

sup{f(s), 0} dν(s) =

ˆ

A

f+(s) dν(s).



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 13

For µ− we proceed similarly. Finally

var(µ)(A) = ssup{µ,−µ}(A) =

ˆ

A

sup{f(s),−f(s)} dν(s) =

ˆ

A

|f(s)| dν(s)

and the proof is complete. �

Example 4.13. Consider f : R → R integrable and define α on (R,B(R)) by

α(A) =

ˆ

A

f(t)dt.

By Corollary 4.12, the total variation of α is given by

var(α)(A) =

ˆ

A

|f(t)| dt.

When applying this with A = (a, x], we can use Riemann-type partitions in the definition of

the supremum (see for instance Remark 2.10 in [9]). We obtain a very direct proof of the

identity

V [F ; a, x] =

ˆ x

a

|f(t)| dt, for F (x) :=

ˆ x

a

f(t)dt.

4.2. Application: Vector measures on a space of bilinear forms. Consider a σ-finite

measure space (S,Σ, ν) and a separable Banach space X . We denote by Bil(X) the space

of bounded bilinear forms defined on X ×X , and by L(X,X∗) the space of bounded linear

operators from X to X∗. In both spaces, we use the σ-algebra of Borel sets. Consider

Q : S → L(X,X∗) Bochner-integrable and define a Bil(X)-valued measure α by

α(A)(x, y) =

ˆ

A

〈Q(s)x, y〉 dν(s), A ∈ Σ, (x, y) ∈ X ×X.

Theorem 4.14. The vector measure α is of bounded variation and its variation is given by

var(α)(A) =

ˆ

A

‖Q(s)‖L(X,X∗) dν(s), A ∈ Σ.

In other words, var(α) << ν and
d

dν
var(α) = ‖Q‖ .

Proof. Notice that
ˆ

A

‖Q(s)‖ dν(s) =

ˆ

A

sup
‖x‖=‖y‖=1

〈Q(s)x, y〉 dν(s),

where the supremum can be computed over (xn, xm), for a dense sequence (xn) on the unit

sphere SX . We do not use absolute value here because, when 〈Q(s)x, y〉 is negative, we can

replace x with −x. We apply Theorem 4.11 with f(x,y) = 〈Q(·)x, y〉. We obtain

ˆ

A

‖Q‖ dν =

(

ssup
‖x‖=‖y‖=1

α(·)(x, y)

)

(A) = sup
Π∈P(A)

∑

C∈Π

sup
‖x‖=‖y‖=1

α(C)(x, y).



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 14

We conclude that

ˆ

A

‖Q‖ dν = sup
Π∈P(A)

∑

C∈Π

‖α(C)‖
Bil

= var(α)(A)

and, since the left-hand side is finite, α is of bounded variation. �

Acknowledgements This work was partially supported by The University of Costa Rica

through the grant “C3019- Cálculo Estocástico en Dimensión Infinita”.

Data Availability. Data sharing not applicable to this article as no datasets were generated

or analysed during the current study.

Declarations

Conflict of interest The authors have no conflicts of interest to declare that are relevant

to the content of this article.

References

[1] Cambronero, S.; Campos, D.; Fonseca-Mora, C.A.; Mena, D: Quadratic Variation for Cylindrical

Martingale-Valued Measures, Preprint.

[2] Diestel, J.; Uhl, J. J., Jr: Vector measures. Mathematical Surveys, No. 15. American Mathematical

Society, Providence, R.I. (1977).

[3] Dinculeanu, N.: Vector integration and stochastic integration in Banach spaces. Pure and Applied

Mathematics. Wiley-Interscience, New York, (2000).

[4] Folland, G.Real analysis. Modern techniques and their applications. Second edition. Pure and Applied

Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York,

(1999).

[5] Kallenberg, O. Foundations of modern probability. Third edition. Probability Theory and Stochastic

Modelling, 99. Springer, Cham, (2021).

[6] Kluvánek, I.: The extension and closure of vector measure. Vector and operator valued measures and

applications (Proc. Sympos., Alta, Utah, 1972), pp. 175–190. Academic Press, New York (1973).

[7] Métivier, M; Pellaumail, J.: Stochastic integration, Probability and Mathematical Statistics, Academic

Press, New York (1980).

[8] Nygaard, O; Põldvere, M.: Families of vector measures of uniformly bounded variation, Arch. Math.

88 (2007).

[9] Veraar, M. ; Yaroslavtsev, I.: Cylindrical continuous martingales and stochastic integration in infinite

dimensions. Electron. J. Probab. 21, Paper No. 59, 53 pp. (2016)

[10] Walsh, John B.: An introduction to stochastic partial differential equations. École d’été de probabilités

de Saint-Flour, XIV—1984, 265–439, Lecture Notes in Math., 1180, Springer, Berlin (1986).



ON THE SUPREMUM OF A FAMILY OF SET FUNCTIONS 15

Centro de Investigación en Matemática Pura y Aplicada, Escuela de Matemática,

Universidad de Costa Rica

Email address : 1 santiago.cambronero@ucr.ac.cr

Email address : 2 josedavid.campos@ucr.ac.cr

Email address : 3 christianandres.fonseca@ucr.ac.cr

Email address : 4 dario.menaarias@ucr.ac.cr


	1. Introduction
	2. The set function supremum
	3. Sub-additive set functions on rings
	4. Supremum of signed measures
	4.1. Signed measures with density
	4.2. Application: Vector measures on a space of bilinear forms

	Declarations
	References