^{1}

^{2}

^{1}

^{2}

We prove that the Burkill-Cesari integral is a value on a subspace of

Since the seminal Aumann and Shapley's book [

To the best of our knowledge, the more general contribution so far on the link between derivatives of set functions and value theory is Mertens [

In Epstein and Marinacci [

In a previous paper [

Motivated by all these facts, in [

In this paper we extend our investigation to develop the connection with the theory of value or of semivalue, also in the light of the problem exposed above. In Section

Unfortunately, the BC integral proves to be not continuous with respect to the

From now on we will denote by

A set function

For the sake of brevity we refer the reader to [

Throughout the paper we will write

A chain

A chain will be identified with the subchain consisting of all the links. Given a game

If

The space

We also refer the reader to [

A

As in [

The BC integral does not depend upon the integration mesh (see Proposition 5.2 in [

As we will see, the space BC contains many games which are of interest in the literature: we begin by recalling a sufficient condition for vector measure games to be in BC, which is an immediate consequence of [

Let

Anyway, the class of BC integrable games is not limited to smooth measure games. Indeed note that the converse implication of the previous proposition does not hold: consider as in [

As for the relation between the spaces

In fact

So

Therefore, we can now consider the space

The following result states that the same measure can be used for the absolute continuity and the BC integrability of a game in

The space

The fact that each game

Conversely, let

From [

Let

The space

We need to prove that for every

Let

Fix

It is also immediate to check that

It remains to prove that

To this aim, for any

According to [

A linear mapping

(

(

(

when

(

The following result immediately derives from (

The mapping

Consider now the space

The next example shows that this space contains several games which do not belong to

Let

Finally

It is clear that one can use functions of different forms to provide classes of measure games

On

Denote

As in the previous example,

Furthermore

Thus

Again take

Finally, the following example shows that

Consider the sequence of scalar measure games

In [

For a scalar measure game

(

To prove that (1) implies (2), assume that

Then, by Lyapunov theorem, there exist sets

Also (3) implies (1) trivially.

It is immediate to note that

Again from [

Let

Let

Consider now the vector subspace

For instance, consider the function

Also the inclusion

Let the scalar measure game

The implication (1)

We turn then to the implication (2)

As

Choose now the following

Then for

We have then, similar to the computation in Proposition

Therefore, for instance, taking

We will need in the sequel the following lemma.

The space

Fix

Let

In a completely analogous way, as

Moreover,

Again fix

Take

Now

Hence

In

The BC integral

Let

Hence for

Finally, we deduce from (

According to Theorem 1.8 in [

Let

If

Because of (

Similarly

In conclusion

Since powers of probabilities belong to

We point out that, as

Moreover from [

Define now the space

Denote by

For the converse inclusion, take

Indeed, by the feasibility of

Obviously

Let

We claim that

The function

Indeed one can provide infinitely many examples of subspaces of

It may anyway be of interest to characterize games in

We will first prove that the measure game in Example

Let

Fix a Hahn decomposition

We will prove that

Fix

In fact, let

Indeed, suppose, for instance, that

Then an easy computation shows that in both cases (either

In a completely analogous way one shows that the game

To prove that

Assume that

We will also use the alternative form

Now by (

Note first that since

Recall now the definition of

In particular, since

We note now that, since

Now, since

But then

The authors declare that there is no conflict of interests regarding the publication of this paper.