On the Differentiability of Vector Valued Additive Set Functions

functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.


Introduction
This note can be considered as a continuation of the work started by the first author in [1]. Throughout the paper is a 2     -ring of subsets of a nonempty set .
is  -Lebesgue-Bochner integrable, then it is easily seen that the function : is a vector measure. It readily follows from the property of the Lebesgue-Bochner integral that whenever Such a property is expressed by saying that  The classical Lebesgue-Nikodým theorem states that for an additive real valued set function (not necessarily countably additive) : F    ,  -absolutely continuity implies  -Lebesgue differentiability (see for example [2]).  [3]).
In this note, we reactualize the approach to the definition of the integral first introduced in [1] in order to obtain a Lebesgue-Nikodým type theorem on the differentiability of Banach space valued additive set function.
The exposition will be organized as follows. In Section 2, we introduce a new approach to integration theory that does not require the elaborate machinery of Lebesgue measure theory, and at the same time significantly sim-plifies the approach to gauge integral. In Section 3, we shall see that the space of classes of integrable functions (in the sense of the definition of integrability in Section 2) can be naturally given a Banach space structure. In Section 4, we state and prove our main result which can be seen as an extended vector valued version of the Lebesgue-Nikodým theorem. The fifth section is devoted to some extension of the Fundamental Theorem of Calculus.

Extended Notion of Integral
We begin by recalling the definition of limit in its most general form, that is, the Moore-Smith limit, also known as the net limit.
A nonempty set is said to be directed by a binary relation , if has the following properties: The notion of convergence can then be defined whenever the set X is a metric space, with distance function .
  a Ordinary sequences n , in which n D   directed by constitute a special case of nets. The net limit takes over all the essential parts of the theory of limits of sequences; to name a few: the uniqueness of limit, the algebraic properties of limits, the monotone convergence property of limits, the Cauchy criterion, and so on. For more details on net limits, we refer the reader to [4].

>,
Next we introduce the notion of size-function that will generalize the usual notion of measure in integral theory. In the following, is a nonempty set. The power set of , that is, the set of all subsets of will be denoted by We denote by the subset of P  A obtained by taking the union of all elements of A .
if we wish to specify the tagging points. We denote by we say that Q is a refinement of P and we write if P Q  P and . P Q 

 
It is readily seen that such a relation does not depend on the tagging points. It is also easy to see that the relation Clearly, and P Q . Thus the relation has the upper bound property on  We then infer that the set is directed by the binary relation .
In what follows, X will denote either a real or a complex normed vector space. Given a function : f X   , and a tagged  -subpartition . For convenience, we are going to denote whether or not the net limit exists.
We are now ready to give the formal definition of the resulting integral. Definition 6. We say that a function : -integrable functions over the set . A Many classical properties of the integral follow immediately from the properties of net limits and therefore their proofs are obtained at no extra cost. To name a few, we have the uniqueness of the integral, the linearity of the integral operator, the Cauchy criterion for integrability, and so on.
It is also important to realize that no notion of measurability nor notion of gauge is postulated in the above definition.
We finish this section by noticing that if 1 2 in then

Spaces of Integrable Functions
In this section, we show that the space can be given a structure of complete seminormed space. We fix a (real or complex) normed vector space X In view of Proposition 3 and Proposition 7, we can always regard integrability only with respect to a size-function η defined on the whole of and we shall write Clearly, a Riemann integrable (resp. Lebesgue integrable) function can be regarded as an integrable function in the sense of Definition 6 with respect to the outer-measure obtained by the Carthéodory extension of the length function defined on the ring of bounded intervals (resp. the Lebesgue measure   defined on a  -algebra of subsets of  ).
It should also be noticed that if the set A is such that then for all subpartitions  , and thus A It follows that the integral does not distinguish between functions which differ only on set of size zero. More precisely, We say that a function f is  -essentially equal on A to another function , g and we write , We shall denote by  

Extended Lebesgue-NikoDým Theorem
For such and there exists P , Q other h o su On the and, it follows fr m (3) that there exists Finally, by definition of the integral, there ex Combining (7), (8), and (9), we have for The desired result follows since > 0  is arbitrary chosen. The proof is complete.
As a direct corollary of our main Theorem 10, we do have the following extended version of the vecto ued Radon-Nikodým Theorem.
Corollary 11. Let be a σ-algebra of a set