1. Introduction
The Vitali convergence theorem [
1], which owes his name to the Italian mathematician Vitali, is a generalization of the Dominated convergence theorem developed by Lebesgue. It is a characterization of the convergence of a sequence
in
in terms of uniform integrability and convergence in the
m-measure of
. Our goal is to show an analog of Vitali’s classic theorem for varying measures, namely when
and
are simultaneously convergent in some sense. In particular, we want to identify sufficient conditions for the following chart to hold, for the different types of convergence for varying measures. Obviously, we will have different versions depending on the assumptions we use on the sequence of measurable functions
and on the varying measures
.
Additionally, the convergence of
to
may be considered in a strong or weak sense, when the functions involved are vector-(multi)valued. Fundamental tools for obtaining results of this type are the “absolute” or “uniform” continuity of integrals. There is a wide literature on the convergence of measures, since it has applications, for example, in probability and statistics, stochastic processes, control and game theories, symmetric diffusion processes, transportation problems, signal and image processing, neural networks, symmetric operators, continuous dependence for measuring differential inclusions and measuring differential equations [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14].
In
Section 2, some results, related to (
1) for the convergence theorems of the type of Fatou, Monotone, Lebesgue or Vitali are highlighted, emphasizing both the hypotheses on the space
, on the sequence of functions
and on the sequence of measures
. There is a large literature on this subject, the first paper on which was published by Serfozo [
15]. This problem was recently addressed by various authors in [
16,
17,
18,
19,
20]. Some of these results are given in
Section 2 and are obtained for sequences of scalar or vector functions when the varying measures are finite and countably additive or probabilities. In [
21,
22], some of these results have been extended to the multivalued case with weakened assumptions, while in [
23] the varying measures considered are only subadditive. Above all, we will highlight the results of these last three quoted papers because they will be the starting point for the new part that is contained in
Section 3. Here, (
1) is considered to have a stronger definition of integrability and results are given, either existing (the compact case) or new (the noncompact case), when the approximating sequence
is integrable in the sense of McShane.
2. The State of the Art
Let be a measurable space with be a -algebra, be the space of measurable functions and let be the vector space of finite real-valued measures on . Let be the cone of non-negative elements of . Let be the total variation of a measure m and be its positive and negative parts, respectively. The symbol denotes the absolute continuity of m with respect to .
Let
X be a Banach space with dual
and
be the unit ball of
. We denote with
the family of all nonempty, convex, weakly compact subsets of
X. For every
, let
, for each
be the
support function of and
is the Hausdorff metric on the hyperspace
in which we consider the Minkowski addition (
) and the standard multiplication by non-negative scalars, see [
24] for other properties.
When working with varying measures different types of convergence can be considered. Concerning the setwise and the convergence in total variation, no more conditions are imposed on the measurable space , while for the weaker convergences a topology is necessary, is supposed to be a locally compact Hausdorff space and is its Borel -algebra (in this case we use the symbol ). In these latter cases, let , , and be the families of all continuous functions, and their subfamilies that vanish at infinity have compact support and are bounded.
Definition 1. Let m and be in , we say that a sequence
- (1a)
Converges vaguely to m () if
- (1b)
Converges weakly to m () if
- (1c)
Converges setwisely to m () if for all or, equivalently, if since simple functions are dense in the space of bounded measurable functions;
- (1d)
Converges in total variation to m () if . Then is convergent to m uniformly on , ([25], Section 2); - (1e)
Is bounded if ;
- (1f)
Is uniformly absolutely continuous with respect to m (), if for every there exists such that when and .
Interesting comparisons among all these definitions are given in [
25,
26,
27]. In general, the setwise convergence is stronger than both the vague and the weak convergences; weak convergence is stronger than vague convergence. Moreover if
is simultaneously vaguely convergent and uniformly absolutely continuous with respect to
m, then it converges weakly to
m, while if
is vaguely convergent to
for every
then it converges setwisely, (see [
22], Remark 2.2 and Proposition 2.3).
If
and
m are measures in
and if the sequences
are setwise convergent to
, respectively, then
is setwise convergent to
m and the reverse implication generally fails, see for example ([
21], Remark 2.2).
Finally, according to ([
28], Corollary 8.1.8 and Remark 8.1.11), if
is an arbitrary completely regular space and let
with
m Radon and
, the convergence (1a) is equivalent to the following (Portmanteau result):
- (1g)
for any closed set , .
Regarding
, for those who are interested we quote [
29] and the references therein.
Definition 2. Let and be a sequence of measurable functions. We say that
Has uniformly absolutely continuous -integrals on Ω (u.a.c.), if for every there exists such that for every with then for every .
Is uniformly -integrable on Ω (u.i.), if
It is obvious that if a sequence
of measurable functions is uniformly bounded, then it is uniformly
-integrable for an arbitrary sequence
such that
. Moreover, for a bounded sequence of measures
, the sequence
is uniformly
-integrable on
if and only if it has uniformly absolutely continuous
-integrals and
([
21], Proposition 2.6). Finally, in [
15], the uniform absolute continuity is given in a partly different form, but Serfozo’s one and (
u.a.c.) are equivalent. Many results in the literature are related to (
u.i.); in this note we put in evidence sufficient conditions that use the weakest condition (
u.a.c.).
All the results of
Section 2.1 and
Section 2.2 resulting from [
21,
22] are given for
unless otherwise specified. The quoted results from other papers are given in general from sequences of probability measures or equibounded sequences of measures.
2.1. The Nontopological Case
In this subsection sufficient conditions are given for the problem (
1) when the sequence of varying measures converges setwisely in an arbitrary space. The setwise convergence is a high power tool since it permits strong results to be obtained. Let
be a measurable space, with
an arbitrary
-algebra. For existing results of type (
1) in the literature, we would like to cite some results of Fatou, Monotone, Dominated or Vitali types:
In [
15], we have Theorems 2.4, 2.7 and 2.8. In the first theorem, a liminf setwise-type convergence is considered to obtain a Dominated convergence theorem under suitable hypotheses on a sequence
that dominates
, while the other two are necessary and sufficient conditions to obtain a Vitali-type theorem and a Lebesgue theorem under tightness and
u.i. conditions. All the results are given for scalar functions.
In [
16], in Theorem 2.2, the authors give a Fatou- and a Lebesgue-type theorem under an inequality of the
on each Borelian set; in particular for the Dominated convergence theorem the sequence
is equibounded by a measure
. In this case, the authors weaken the setwise convergence, but need a topology on the space and the Borel
-algebra. Both results are given scalar functions.
In [
20], Theorem 4.2, Corollaries 5.3 and 6.2 give Fatou-, Lebesgue- and Monotone-type results for the setwise convergence of a sequence of probability measures, respectively. All the results are given for scalar functions.
In [
21], first the authors consider the case of scalar integrands and obtain the following result for finite, non-negative measures using the uniform absolute continuity of the integrals and the setwise converges for the measures.
Theorem 1 ([
21], Theorem 2.11)
. Let . Suppose that- (1i)
, in m-measure;
- (1ii)
satisfy (u.a.c.);
- (1iii)
.
Then and for all , An analogous result follows for signed measures when the convergence of
is in
-measure and (
u.a.c.) is considered with respect to
, ([
21], Corollary 2.13).
Moreover in [
20], Corollary 5.3, a similar result is obtained with different hypotheses, which implies the equiboundedness of the sequence
which is not required in Theorem 1.
A simple example of application of the previous theorem can be the following:
Example 1. Let
,
the Borel
-algebra,
the Lebesgue measure and
, or
for every
.
and
. For every
let
. We divide each interval
into
n pairwise disjoint equi-measurable intervals,
of
-measure
. For every
let
. Let
the sequence of functions defined as follows:
obtained by the construction
pointwise and then in
-measure. Moreover, the sequence
is (
u.i.) with respect to
and so it is (
u.i.) with respect to
. By [
21], Proposition 2.6, the sequence
is (
u.a.c.). Then, the pair
satisfies Theorem 1, but the sequence
is not dominated by an integrable function since, if such a function exists, then it will dominate
.
For the multivalued case we recall that any map
is said to be
scalarly measurable if for each
is measurable the scalar function
; given a measure
m,
is said to be
scalarly integrable with respect to m if for each
is integrable the scalar function
and
is said to be
Pettis integrable in
with respect to
m if
is scalarly
m-integrable and for every
,
exists so that
and in such a case
. We denote by
the class of all scalarly
m-integrable and
-valued multifunctions which are Pettis
m-integrable. Similarly, we write
for vector-valued functions. For results concerning the Pettis integrability of vector-(multi)valued functions, see for example [
24,
30,
31,
32,
33,
34,
35,
36,
37].
The (u.a.c.) takes the following form for sequences of multifunctions: has uniformly absolute continuous scalar -integrals on (u.a.c.s.), if,
For every
, a positive
exists so that for every
and
it is:
A first result of the multivalued case is the following:
Theorem 2 ([
21], Theorem 3.2)
. Let be scalarly m-integrable, and for every , let be in . Then if- (2i)
satisfies (u.a.c.s.);
- (2ii)
;
- (2iii)
for all and for all
To prove the Pettis
m-integrability of
, it is sufficient to show that it is determined by a weakly countably generated subspace of
X ([
35], Theorem 2.5). As a consequence of Theorem 2, we have
Theorem 3 ([
21], Theorem 3.3)
. Let be scalarly measurable and for every let be in . If- (3i)
For each , converges in m-measure to ;
- (3ii)
Γ and satisfy (u.a.c.s.);
- (3iii)
,
then and for all and for all Moreover, if the convergence in m-measure is strengthened with a scalar equiconvergence in measure with respect to the sequence and the setwise convergence is substituted with the convergence in variation, a stronger result is obtained.
Theorem 4 ([
21], Theorem 3.5)
. Let be scalarly measurable and for every let be in . If- (4i)
For all ,
- (4ii)
Γ and satisfy (u.a.c.s.);
- (4iii)
Γ has uniformly absolutely continuous scalar m integral;
- (4iv)
,
then and uniformly in These results can be applied directly to the vector-valued case. Indeed, in the vector case
for every
. The corresponding results are [
21], Theorem 3.7 and [
21], Theorem 3.9. Therefore we can deduce the vector case directly from multivalued case. Here, we report, as an example, the vector-valued formulation of Theorem 3 for which in [
21], Theorem 3.7, a direct proof is also given that makes use of a Grothendieck characterization of weakly compact sets.
Corollary 1 ([
21], Theorem 3.7)
. Let be in . If- (1j)
, in m-measure for each ;
- (1jj)
satisfy (u.a.c.);
- (1jjj)
,
then and, weakly in X, we have 2.2. The Topological Case
Sometimes in applications it is difficult, at least technically, to prove that the sequence of measures converges for every measurable set. Therefore, other types of convergence could be considered, based on the structure of the topological space
. Following [
28], we assume that
is only an arbitrary locally compact Hausdorff space and
is its Borel
-algebra. All the measures we will consider on
are finite. Moreover a measure
m is Radon if it is inner regular in the sense of approximation by compact sets. For existing results of type (
1) in the literature, in this setting, we quote, for example,
The paper [
15] (Theorems 3.3 and 3.5) in locally compact second countable and Hausdorff spaces, when the sequence of scalar functions
converges continuously to
f, for vague and weak convergences, respectively. Under a domination condition with a suitable sequence
in the first result while, in the other, the uniform
-integrability of the sequence
, with
for every
, together with a condition for
, a Lebesgue’s result is given. In [
15], Lemma 3.2, a Fatou’s result is obtained.
The paper [
16] (
Section 3), where a Monotone convergence result is obtained for locally compact separable metric spaces requiring weak convergence of the sequence of measures and that the space
X is a Banach lattice.
The papers [
19,
20], where Fatou’s, Monotone and Dominated convergence results are obtained in metric spaces, for sequences of scalar lower semi (equi)continuous functions
satisfying an asymptotic uniform
-integrability, when the sequence of measures converges weakly.
In [
22], a Vitali result in the scalar case is obtained:
Theorem 5 ([
22], Theorem 3.4)
. Let m be a Radon measure and let . Suppose that- (5i)
, m-a.e. with ;
- (5ii)
and f satisfy (u.a.c.);
- (5iii)
and .
Then, and for every The result of this theorem is still valid if we replace convergence
m almost everywhere with convergence in the
m measure. The formula (
3) relies on the Urisohn’s and a Portmanteau Lemma and it was first proven for arbitrary compact sets and finally for Borelian sets, since
m is a Radon measure. This allows us to deduce the setwise convergence of
, considering
for every
([
22], Corollary 3.5).
Using the scalar case and the support functions, an analogous result is obtained for the multivalued case.
Theorem 6 ([
22], Theorem 4.2)
. Let m be a Radon measure and Γ
be a scalarly continuous multifunction and for every . If- (6i)
and Γ satisfy (u.a.c.s.);
- (6ii)
m-a.e. for each ;
- (6iii)
and ,
then and, for every and , it is We observe in the previous theorem that the null set in (6ii) may depend on .
If we assume by hypothesis
We can remove the assumption on and the convergence of the sequence and we can obtain the next result in a general measure space without any topology.
Theorem 7 ([
22], Proposition 4.4)
. Let Γ
be a scalarly m-integrable multifunction and for every . Suppose that- (7i)
satisfies (u.a.c.s.);
- (7ii)
;
- (7iii)
for every and it is
Then .
Recently, another paper was published on this subject [
38], for the weak convergence of integrals, referred to the vector values case, when
is a metric space,
X is a complete paranormed vector spaces and the sequence of probability measures converges weakly. Therefore, in this case, the target space
X is more general, but the sequence of measures is equibounded.
2.3. The Nonadditive Case
Additionally the study of nonadditive measures has played an important role because of its applications in probability, statistics and in all applied sciences where uncertainty must be considered. Therefore, in this case, we quote a Monotone convergence result of type (
1). Let
be a locally compact Hausdorff space,
the family of all subsets of
and
be a
-algebra of subsets of
. We denote with the symbol
the family of bounded, real valued functions. We begin by recalling the scalar nonadditive measures: let
be a
a submeasure (in the sense of Drewnowski [
39]), namely with
monotone (if
, for every
, with
) and
subadditive (if
for every disjoint sets
). Let
be the variation of
m defined by
for every
,
m is said to be
of finite variation (on
) if
. For the properties of
, we refer for example to [
40,
41,
42].
Definition 3. A sequence of submeasures setwise converges to a submeasure m if, for every ,
Since
for every
, the convergence given in Definition 3 is the (1c); the converse does not hold in general. Nevertheless, from [
43], Remark 1, in the countable additive case the two definitions coincide, so we use the same notation.
We denote by the symbol
the family of all nonempty convex compact subsets of
. We consider on
the weak interval order:
if and only if
and
and a multiplication
. For what concerns the (lattice) weak order ⪯ and for its meaning and uses, we refer to [
44].
Definition 4. Let . M is an interval valued multisubmeasure if
- (4a)
- (4b)
for every with (monotonicity);
- (4c)
for every disjoint sets (subadditivity).
In literature the monotone multimeasures that satisfy
are also called set valued fuzzy measures. For the results on this subject, see for example [
40,
42,
45].
Remark 1. Given two submeasures with for every let be defined by According to [42], Remark 3.6, M is a multisubmeasure with respect to the weak interval order ⪯ if and only if are submeasures. Moreover M is monotone or finitely additive if and only if the set functions and are the same (see [41], Proposition 2.5, Remark 3.3). Moreover, following [23], Definition 2.2, setwise converges to if and only if setwise converges to for . In this framework we consider the Riemann–Lebesgue integrability studied in [
43,
46]. If
P and
are two partitions of
, then
is
finer than P (
), if every set of
is included in some set of
P. All the partitions we consider in this subsection are countable.
Definition 5. Let be a function and be a set function. f is Riemann–Lebesgue () μ-integrable (on ) if exists such that for every a partition of exists so that for every partition of with , f is bounded on every , with and for all , , is convergent and b is called the Riemann-Lebesgue -integral of f on and is denoted by .
Analogously we can define the Riemann–Lebesgue integrability for the interval multifunctions
, see for example ([
12], Definition 6).
Remark 2. Given an interval multifunction , with and for all , for every tagged partition of , we have that Moreover, according to [
12], Proposition 2,
G it is
integrable with respect to
M on
if and only if
are
integrable with respect to
,
and
Therefore, for every submeasure which is a selection of M and every with then .
Theorem 8 ([
23], Theorem 4.2)
. Let be a sequence of bounded interval valued multifunctions and be multisubmeasures. Suppose that there exist an interval valued multisubmeasure with of bounded variation and a bounded multifunction such that- (8i)
for every and uniformly on Ω;
- (8ii)
for every and setwise converges to M,
3. McShane Integrability
In
Section 2.1 and
Section 2.2, the vector valued case was derived from the multivalued one when we consider Pettis integrability. In the case of the McShane integral it is possible to proceed in the same way. However, thanks to the Rådström embedding, for the McShane integral it is possible to deduce at once the multivalued case from the vector valued case. We recall that the Rådström embedding is a map
that is additive, isometric and positively homogeneous, see for example [
24,
47]. Therefore, if
are McShane integrable multifuctions, then the vector-valued functions
are and viceversa. So the assumptions for multifunctions can be shifted to the corresponding
allowing the convergence to be obtained from the vector case. In general, the Rådström embedding of a Pettis integrable multifunction could not be Pettis integrable. If we ask for a stronger notion of integrability, also under setwise convergence, some topological conditions are needed (see for example [
48,
49,
50,
51]).
3.1. The Compact Case
Let be a compact measure space with a topology . A McShane partition of is a family such that is a finite disjoint cover of by elements of and . A gauge on is a function such that for every it is . A McShane partition is subordinated to a gauge if for .
Definition 6. is said to be -integrable on Ω with - integral if for every a gauge Δ exists such that for each partition subordinated to Δ, we have In this case, we set .
For this kind of integration see for example [
48,
52,
53,
54].
Definition 7. A sequence of -integrable functions is -equi-integrable on Ω
, if for every there exists a gauge Δ
such thatfor each partition subordinated to Δ
and every . We can observe that, thanks to [
48], Theorem 1N, the inequality (
4) holds for every
, as highlighted in [
21], Theorem 3.11. Therefore, we have
Theorem 9 ([
21], Theorem 3.11)
. Let , , and let . If- (9i)
is -equi-integrable on Ω;
- (9ii)
, for all ;
- (9iii)
,
then f is -integrable and, for all , Moreover if condition (9iii)
is replaced with the convergence in total variation (), then (
5)
holds uniformly in . The multivalued case [
21], Theorem 3.13, follows using the Rådström embedding.
3.2. The Noncompact Case
We now want to extend the previous results for the compact case to a more general case, we limit ourselves to the vector case because the multivalued case follows similarly.
Let now
be a nonempty, locally compact Hausdorff space and let
m be a quasi-Radon and outer regular measure. Let
be the subset of
consisting of the outer regular and quasi-Radon measures. A series
exists unconditionally for the norm topology of the Banach space
X if and only if for every
a finite set
exists such that for every finite set
I with
we have
We say that a sequence of series
is
uniformly unconditionally convergent if for every
a finite set
exists so that for every
, we have
A generalized McShane partition of is a family such that is a sequence of disjoint measurable sets of finite measure such that of and for each . A gauge on is a function such that for every . We say that a generalized McShane partition is subordinated to a gauge if for all .
Definition 8 ([
48], Definition 1A)
.A function is said to be m-generalized McShane (-integrable on Ω
with -integral if for every a gauge Δ
exists such that for each generalized McShane partition subordinated to Δ
, we haveWe set .
Remark 3. By [48], Theorem 1N, we know that a function f is -integrable if and only if is -integrable for every subset and In this case we can set for every . Moreover, by [48], Remark of Theorem 1N, if E is such that then Finally, according to [
48], Corollary 2D,
f is
- integrable if and only if for every
there exists a gauge
such that: for each generalized McShane partition
subordinated to
,
exists unconditionally for the norm topology of
X and it is
We begin with a convergence results involving the
-integrability of the limit function
f. To obtain this we need the definition of equi-integrability for series, so we extend (
4) in the following form:
Definition 9. Let , be measures in . We say that a sequence of -integrable functions is -g-equi-integrable on Ω
, if for every a gauge Δ
exists so that for each generalized McShane partition of Ω
subordinated to Δ
and for every the sequence of series is uniformly unconditionally convergent and Remark 4. An immediate consequence of Definition 9 and of [48], Theorem 1N, is the -g-equi-integrability of the sequence with respect to every . In fact it is enough to take for every . In particular if is a compact Radon measure space this is exactly [21], Definition 3.10, since the McShane integrability is given in terms of finite partitions as in Definition 7, so the concept of -equi-integrability on compact sets coincides with that of -g-equi-integrability. If for every , this is the classical condition of equi-integrability. On noncompact sets, the two definitions differ, being the definition of
-g-equi-integrability stronger than that of
-equi-integrability. In particular, if
,
compact for every
, and
is
-g-equi-integrable on
, then
is
-equi-integrable on
for each
. For the converse implication, we can observe that it is false in general. Let
and
. Take
and let
be the Lebesgue measure. Then the collection
restricted to a separate
consists of a finite family, since
on
for every
. Consequently it is equi-integrable on each
. However, it is not g-equi-integrable on
because, otherwise, by [
55], Theorem 4,
will be integrable.
Theorem 10. Let m and be measures in , let be -integrable functions, . Suppose that
- (10i)
is -g-equi-integrable on Ω;
- (10ii)
converges pointwise to a -integrable function f;
- (10iii)
.
Moreover, if we substitute condition (10iii)
with the convergence in total variation (), then (
11)
holds uniformly in . Proof. Let
be fixed. Let
be a gauge on
satisfying (10i), then for every generalized McShane partition
subordinated to
and for every
the sequence of series
is uniformly unconditionally convergent and
Without loss of generality, we may assume that corresponding to
the same gauge
works for the function
f that is
-integrable. Now, let
be a fixed generalized McShane partition subordinated to
. We will now show that the sequence
is Cauchy. We fix
. Since the sequence of the series
is uniformly unconditionally convergent, let
be a finite subset of
such that, for every finite set
I with
and for each
,
We fix
with
I finite. The pointwise convergence of
to
f and the setwise convergence of
to
m implies that
Corresponding to
choose
so that if
Then, if
, by (
14) and (
16)
Then the sequence in (
13) is Cauchy.
Additionally, the sequence
is Cauchy. Therefore, corresponding to
, let
be a generalized McShane partition subordinated to
related to the condition (10i). Therefore, by (
12) and taking into account that (
13) is Cauchy, for
being suitably large we have
which also shows that the sequence
is Cauchy, therefore it converges to
. Let
be an integer such that, for every
We will show that
is the
-integral of
f. Since
is unconditionally convergent let
be a finite set such that
Moreover, since
are uniformly unconditionally convergent without loss of generality, we may assume that for the same set
I the inequality (
14) holds for
. Therefore, if
n is suitably large we have
Therefore it follows that
Finally, if
,
does not depend on
E. Then, formula (
15) holds uniformly on
and the convergence in formula (
11) is uniform. □
Remark 5. Theorem 10 is still valid if we replace the pointwise convergence of to f in condition (10ii) with the convergence in m-measure to f. In fact, by [48], Remark of Theorem 1N, we can pass to the convergence m-a.e., and from this, the theorem holds for every subsequence of . This implies that the result of this theorem is still valid for convergence in the m-measure because there would be a contradiction if, absurdly, a subsequence existed in which it is not valid. In this theorem, the -integrability of the limit function f plays a fundamental role. To weaken this hypothesis and obtain the -integrability as a thesis, we must introduce additional assumptions on . Let be a locally compact second countable Haudorff space (lcsc). Namely,
From now on we suppose that is a sequence of measures in and such that has (lcsc). Thanks to Remark 4 we can apply Theorem 9 to each compact set . Therefore, we have
Corollary 2. Let m and be measures in , with m Radon and Ω that has (lcsc) . Let be -integrable functions, . Suppose that
- (2j)
The sequence is -g-equi-integrable on Ω;
- (2jj)
The sequence converges pointwise to a Pettis m-integrable function f;
- (2jjj)
.
Then, f is -integrable on Ω
and Proof. Let
be fixed. Let
be as in (
lcsc). Since
is relatively compact, we can apply Theorem 9 ([
21], Theorem 3.11) and we obtain that
f is
-integrable on
and there exists
such that
for every
. Then, by [
48], Corollary 4.B,
f is
-integrable on
, since it is Pettis
m-integrable in
and
-integrable in each
. We now apply Theorem 10. □
Remark 6. Moreover, if we substitute condition (2iii) with the convergence in total variation (), then (17) holds uniformly in . Finally, as in Remark 5, the result is still valid for convergence in the m-measure of to f. 5. Conclusions
This paper describes sufficient conditions ensuring convergence in both a weak and strong sense of a sequence for scalar and vector or multivalued Pettis integrable functions when the sequence converges in some sense to a measure m. When convergence is setwise or in total variation, our ambient space is a general measure space, while in the case of vague convergence, the ambient space is a locally compact Hausdorff measure space. Both the case of countably additive measures and that of fuzzy measures has been exposed, together with a comparison with results known in the literature.
When we consider the McShane integrability we can pass from the vector case to the multivalued one. In this setting, we need a topology on the space; thus, is a Radon measure space with a topology and the Rådström embedding is the key to passing from a sequence of McShane integrable functions to a sequence of McShane integrable multifunctions. In this case, new results of Vitali type were also provided. For what concerns future research directions on this subject, we are studying the case of convergence results for fuzzy varying measures that converge weakly; this research is in progress.