Compactness criteria for Stieltjes function spaces and applications

In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli-Arzel\`a and Kolmogorov-Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions.


Introduction
Stieltjes differential calculus is a powerful tool in the study of complex differential problems [13].
It is based on a notion of derivative with respect to a left continuous and non-decreasing function that we call a derivator -see Definition 2.6.This theory has been proven particularly useful while modeling processes in which the features studied have sudden changes in time -see, for instance [6] and [15].
On the other hand, compactness results are crucial when studying the existence of solutions of differential problems (for example, if we want to use fixed point index methods).The first of this kind of results for the case of Stieltjes function spaces appeared in [6,Theorem 4.2], where a sufficient condition of relative compactness in C g ([a, b], ) was given.This same result was deepened (in a slightly more general setting) in [7], giving a characterization for compactness in BC g ([a, b], ) (Theorem 3.4).Nonetheless, the theory is still young, with many classical results without a counterpart in this setting and various open questions unanswered; among them, for instance, if the -rather unwieldy-condition of g-stability -Definition 3.3-can be restated in a more meaningful way, and whether it can be related to the uniform g-equicontinuity.

Preliminaries
Throughout the article, will be or and g : → will be a non-decreasing and left continuous function, which we will refer to as derivator.For a given derivator, we define the sets: C g := {t ∈ : g is constant on (t − ǫ, t + ǫ) for some ǫ > 0}, where ∆g(t) = g(t + ) − g(t), and g(t + ) is the right hand side limit of g at t. C g is open in the usual topology, so it can be expressed as where Λ is countable and the union is disjoint.With this in mind, we also consider the sets By µ g we will denote the Stieltjes measure associated to g (see, for example, [1] for details).A set or a function is g-measurable if it is measurable with respect to the complete σ-algebra obtained through Carathéodory's extention theorem [1,Theorem 1.3.3].We denote this σ-algebra by g .L 1 g (X ; ) is the set of equivalence classes of µ g -integrable functions on a g-measurable set X with values in where f ≡ h if and only if µ g (( f − h) −1 ( \{0})) = 0, i.e., f = h µ g -a.e.Given f ∈ L 1 g (X ; ), we denote its integral by As usual, we do not make the difference between the equivalence classes in L 1 g (X ; ) and their representatives.Remark 2.2.For an arbitrary g, it is not always the case that BC g ([a, b], ) = C g ([a, b], ).That is, g-continuous functions on compact intervals are not necessarily bounded (see [15, Example 3.19]).
The following result describes properties satisfied by g-continuous functions.• f is continuous from the left at every t ∈ (a, b]; • if g is continuous at t ∈ [a, b), then so is f ; The concept of g-continuity is closely related to g-topologies, which we now define.Definition 2.4.For a derivator g : → , the topology induced on the real line by the pseudometric ρ(x, y) = |g(x) − g( y)| is called g-topology, and we denote it by τ g .
Remark 2.5.Analogously to the standard case, it can be seen that g-continuity of f : → is equivalent to the continuity of the map between topological spaces f : ( , τ g ) → ( , τ usual ).
Next we define the central object of Siteltjes differential calculus.
where a n and b n are as in (2.1), provided the corresponding limit exists, in which case we say that f is g-differentiable at t.
We introduce now the concept of absolutely g-continuous function, along with a version of the fundamental theorem of calculus for the g-derivative.This is one of the pillars of the theory.1. F is abolutely g-continuous on [a, b], that is, for every ǫ > 0 there exists δ > 0 such that for any family of pairwise disjoint subintervals Lastly, we review some facts about compactness in metric spaces.
Definition 2.8.Let (X , τ) be a topological space.We say that U ⊂ X is relatively compact if U (the closure of U) is compact.
Definition 2.9.Let (M , d) be a metric space.U ⊂ M is totally bounded if for every ǫ > 0 there exists a finite set Instead of relative compactness, we will often check total boundedness.The following proposition establishes that these are equivalent for complete metric spaces.

Compactness in the space BC g ([a, b], )
An incorrect proof of the second countability of g-topologies appears in [6].The mistake was pointed out in [15], but without giving an alternative proof.That is the first result of the section.
is a base of τ g .g-topologies are therefore second countable, and also τ g ⊂ ( ) (the Borel σ-algebra of ).
With either Sorgenfrey's topology [10, p. 79] or the usual one, is hereditarily Lindelöf, so both unions admit a countable subcover.That is, where I ⊂ and J ⊂ are countable.U is then the countable union of sets that are themselves the countable union (by what was shown at the beginning) of elements in , and we are done.
The characterization of compact sets in the space BC g ([a, b], ) was studied in [7] (although in [7] the functions are assumed to take real values, that does not change the validity of the proofs).In order to present it we need the following definitions.
) is said to be g-stable if, for every t ∈ [a, b) ∩ D g and every ǫ ∈ + , there exist δ ∈ + and a finite covering With these definitions we get the following result.
Furthermore, we can simplify the hypotheses in Theorem 3.4 by observing the following.
Therefore, we have the following corollary.

S(t) is bounded for all t ∈ I and 2. S is uniformly g-equicontinuous,
then S is precompact.[7] leaves open the question of whether this corollary is really an equivalence, that is, if gequicontinuity and g-stability imply uniform g-equicontinuity, which would make the statement in Lemma (3.5) an equivalence.We proceed to settle this issue with the following result.Proposition 3.7.Let F ⊂ BC g ([a, b], ) be g-equicontinuous.Then the following statements are equivalent: 1. F is uniformly g-equicontinuous.
2. For every t ∈ [a, b) ∩ D g and f ∈ F , there exists lim s→t + f (s) ∈ uniformly on F .
Proof.(1) ⇒ (2) Suppose in the first place that the second statement does not hold.Then, for some g is increasing, and therefore regulated, so |g(x n ) − g( y n )| → 0 when n → ∞.Consequently, F can not be uniformly g-equicontinuous.
Thanks to the continuity of g at t, there also exists η(δ t ) > 0 with |g(s) − g(t)| < δ t for every s ∈ (t − η(δ t ), t + η(δ t )).Define is an open cover of ([a, b], τ u ), so we can obtain a finite subcover {I t i } n i=1 ∪ {I t j } m j=1 , where t i ∈ [a, b]\D g and t j ∈ [a, b] ∩ D g .We can assume, discarding redundant intervals, that none of them is contained in the union of the rest, and that they are ordered so that Now for an interval I, we will denote by V (g, I) the total variation of g on I. Consider We have that 0 < δ < ∞ as g is non decreasing and, thus, By the definition of δ, there can be no t i ∈ D g in the interval [t, s).There exist I i 1 , ..., I i n satisfying (3.2) such that t ∈ I i 1 y s ∈ I i n ; we can also suppose, discarding intervals from the beginning and the end, that I i 2 , ..., I i n−1 ⊂ (s, t) (the remaining cases are simpler, because they involve only one or two intervals).Since |g(s) − g(t)| < δ, again by the definition of δ, V (g, I i 2 ) = • • • = V (g, I i n−1 ) = 0 and thanks to (3.2), g is constant on n−1 k=2 I i k .g-continuity implies that f is also constant on I i j for every j = 2, ..., n − 1 -see Proposition 2.3, so if we choose t i j ,i j+1 ∈ I i j ∩ I i j+1 for each j = 1, ..., n − 1, we have that Corollary 3.8 has interesting applications.Among them, it offers a way to prove a version of the Weierstrass Approximation Theorem for the space BUC g ( , ) of bounded uniformly gcontinuous functions with the supremum norm.Given that the uniform limit of uniformly gcontinuous functions is a uniformly g-continuous function (this is due to Corollary 3.8 and the fact that the uniform limit of g-continuous functions is g-continuous [6, Theorem 3.4] and the uniform limit of regulated functions is regulated), BUC g ( , ) is a Banach subspace of the Banach space of bounded g-continuous functions with the supremum norm BC g ( , ).
Proof.Those two conditions implying precompactness is precisely the statement of Corollary 3.6 (notice that since S is uniformly g-equicontinuous, we have in particular that S ⊂ BUC g ([a, b])).
For the other direction, Theorem 3.4 tells us that S(t) is bounded for each t ∈ [a, b], and that S is g-equicontinuous.Since S is a family of regulated functions precompact in the supremum norm topology, [11, Theorem 1] tells us that the right hand side limits at the points of D g exist uniformly on S. Applying Proposition 3.7, S is uniformly g-equicontinuous.
Furthermore, since f is uniformly g-continuous, it is regulated by Corollary 3.8.σ is increasing, so the composition f •σ = h is regulated as well.Observe that \g( ) has no isolated points because g is non-decreasing and, therefore, the connected components of \g( ) are nondegenerate intervals.In fact, we can write in the case g is bounded, jump points are not followed by constancy intervals and g does not reach a maximum or minimum, with similar expressions (the inclusion of the endpoints of the intervals may vary) depending on other conditions such as whether g is unbounded from above or below and the placement of its intervals of constancy.This fact implies that h can be continuously extended, in a unique way, to a uniformly continuous function h defined on g( ), as the lateral limits of h exist at every point of g( ) where they can be considered due to the fact that it is regulated.Finally, g( ) is closed, so its complement is open (in the usual topology) and \g( ) = and the intervals (c n , d n ) are pairwise disjoint.For instance, in the same situation as above, we would have We define f is clearly uniformly continuous by construction (as it is defined by saving the 'gaps' in the graph of h with straight lines joining the endpoints) and f • g = f .
Remark 3.14.The construction of f from h could also be carried out by adequate versions of the Tietze extension theorem for uniformly continuous functions on metric spaces [14].Observe that, contrary to the classical case, not all g-continuous functions are locally bounded locally uniformly g-continuous functions [6,Example 3.3].On the other hand, by Corollary 3.8, a function is locally bounded locally uniformly g-continuous function if and only if it is g-continuous and regulated.
Let us denote by [g] the ring of polynomials on the variable g, that is, the set of all functions of the form n k=0 a k g k where a k ∈ , k = 1, . . ., n and n ∈ .For the following result we will restrict the functions in ).By Theorem 3.13 and Remark 3.
for some uniformly continuous function f : By the Weierstrass Approximation Theorem, there exists a polynomial p ) comes with the important caveat that the functions in [g] are not g-smooth due to the product rule [4].Even so, as we can conclude, as an straightforward consequence, that BUC([a, b], ) is a separable Banach space.
4 Compactness in the space L p g ( , ) In this section we prove an extension of Kolmogorov's compactness theorem for L p spaces.Before recalling the classical results -Theorems 4.9 and 4.10), let us start with some technical results from [5] that allow us to decompose Stieltjes integrals.

Definition 4.1 (Continuous and jump parts).
For a derivator g, we define g B : → as which is left continuous and increasing.So is the mapping g C : → given by We refer to g C and g B as the continuous and jump parts of g, respectively.

Lemma 4.2 ([5, Lemma 2.2]
). g, g C and g B satisfy the following properties: Now we define a function similar to σ occurring in the proof of Theorem 3.17.

Definition 4.3 (Pseudoinverse of g C ).
For a derivator g : → whose continuous part is not constant on intervals of the form (−∞, t], the pseudoinverse of g C is the function Proposition 4.4 ([5, Proposition 2.6]).The pseudoinverse of g C satisfies the following properties: • For every t ∈ , γ(g C (t)) t.
• γ is left continuous, and continuous in x ∈ g C ( )\g C (C g ).
Lemma 4.5.For any f ∈ L 1 g ( , ), Proof.In [5, Lemma 2.3], this result is proven for a bounded interval [a, b).Applying this case and the dominated convergence theorem to f n = χ [−n,n) f , the result for follows.
The following result was also presented originally for a bounded interval.
• The pseudoinverse of the continuous part γ : , where µ is the Lebesgue measure.
The proof of the following classical results regarding compactness in L p spaces can be found in [ for every ǫ > 0 there exists ρ > 0 such that for every f ∈ and h ∈ with |h| < ρ, Remark 4.11.The Theorem 4.10 is often stated with the additional condition that is a bounded subset of L p ( , ), but it is possible to prove that this is implied by conditions 1 and 2 -see [9].
Before moving on to the main result of the section, we need to make one last consideration.The pseudoinverse of g C , γ, is defined on the set g C ( ).Consequently, for a family ⊂ L p g ( , ), ).Since we can embed this space in L p ( ) (defining its functions as zero outside of g C ( )), we will treat • γ as a subset of L p ( , ).Theorem 4.12.Suppose that 1 p < ∞ and D g = {d 1 , d 2 , ...}.A subset of L p g ( , ) is totally bounded if and only if for every ǫ > 0 there exists n such that for any f ∈ , 4. for every ǫ > 0 there exists ρ > 0 such that for every f ∈ and h ∈ with |h| < ρ, , thanks to Corollary 4.8, we have that Suppose that ⊂ L p g ( , ) is totally bounded.With the previous equality in mind, it is clear that an ǫ-net of induces an ǫ-net on the sets which are then totally bounded.Using Theorems 4.9 and 4.10, we conclude that satisfies conditions 1-4 (for condition 1, note that for each fixed Let us finally check that if all of the conditions are satisfied, every sequence in has a Cauchy subsequence, which is equivalent to being totally bounded.Take satisfy the hypotheses of Theorems 4.9 and 4.10, respectively, so { f n • γ} n∈ admits a Cauchy subsequence { f n k • γ} k∈ in L p ( , ) and {( f n k (d j )∆g(d j ) 1/p ) j∈ } k∈ also admits a Cauchy subsequence {( Remark 4.13.As in Theorem 4.10, conditions 3 and 4 imply that • γ is a bounded subset of L p ( , ).

An application to Stieltjes-Sobolev spaces
In this section we define the Stieltjes-Sobolev spaces and generalize the classical results concerning continuous and compact inclusions of Sobolev space theory.In particular, these compactness results will be useful when it comes to studying nonlinear Stieltjes differential equations.
When integrating funtions, we will use the following notation for convenience: We start by defining the concept of Stieltjes-Sobolev space W 1,p g (I, ).
We endow W 1,p g (I, ) with the norm and, we will write, in general, W 1,p g (I) ≡ W 1,p g (I, ) and follow the same convention for the rest of spaces.
In the case b = ∞, we take x ∈ [a, b).
Remark 5.4.In the particular case a, b ∈ , a < b, since Therefore, the elements of the space W 1,p g ([a, b)) are g-absolutely continuous functions whose g-derivatives are in the space In the next lemma we will further show that the embedding )), we define: Therefore, given 1 p ∞, there exists a positive constant Thus, there is a constant, which we will continue to denote in the same way, such that Now, we have that u(t) = v(t) + u(a), in particular u(a) = u(t) − v(t), so there exists another positive constant, that we will continue to denote in the same way, such that Finally, Theorem 5.6.Let I ⊂ be an interval which is closed from the left (in the case it is bounded from below) and open from the right.We have that W where C ≡ C(p, µ g ([x, y))) > 0 is a positive constant.Thus, because of inequality (5.1) and the fact that lim n→∞ u n − u L p g (I ) = 0, we have that, for x, y ∈ I, x < y, Thus, u n converges to u in W 1,p g (I).Now, given p ∈ (1, ∞), we have by [12,Theorem B.92] that L p g (I) is reflexive.Thus the product space W = L p g (I) × L p g (I) is also reflexive.The operator T : u ∈ W 1,p g (I) → T (u) = (u, u) ∈ W is an isometry from W 1,p g (I) to W . Since W 1,p g (I) is a Banach space, T (W 1,p g (I)) is a closed subspace of W .It now follows that T (W 1,p g (I)) is reflexive, so W 1,p g (I) is also reflexive.Finally, since L p g (I) is separable [2, Theorem 4.13] for p ∈ [1, ∞), we have that W = L p g (I) × L p g (I) is also separable and, therefore, T (W 1,p g (I)) is separable, implying that W 1,p g (I) is separable.

Compact embedding into BC g ([a, b])
One of the classical results in Sobolev spaces is the compact embedding of In the following Theorem we generalize the previous result to the case of Stieltjes-Sobolev spaces, that is, we will prove that the embedding 1}.
To show that the embedding of ) is compact it is enough to show that is a relatively compact subset of BC g ([a, b]).Now, the proof is a direct consequence of Corollary 3.6.Indeed, first, the set {u(t) : Second, is uniformly g-equicontinuous.Given t, s ∈ [a, b] with t s and u ∈ , where p ′ = 1 if p = ∞ and 1/p +1/p ′ = 1 otherwise.Therefore, given ǫ > 0, we can take δ = ǫ p ′ and, then, Remark 5.8.Observe that, thanks to Proposition 3.7, we can replace the condition in the definition of in the proof of Theorem 5.7 by asking for the family to have uniform right-hand side limits in [a, b) ∩ D g .To prove this equivalent condition it suffices to observe that given an element t 0 ∈ [a, b) ∩ D g , we have that Now, by definition of the right-hand side limit at t 0 , there exists an element δ > 0, such that if t ∈ (t 0 , t 0 + δ), then g(t) − g(t + 0 ) < ǫ p ′ .Then, taking δ = ǫ p ′ we have the desired result.

Compact embedding into L q g ([a, b))
In this section we generalize the compact embedding of W 1,1 (a, b) into L q (a, b) for all q ∈ [1, ∞) [2, Theorem 8.8] to the general case of Stieltjes-Sobolev spaces by using Theorem 4.12.We must emphasize that the classical proof of this result goes through the construction of an extension operator P : W 1,1 (a, b) → W 1,1 ( ), for which reflexion and prolongation techniques are used.
In the case of the Stieltjes derivative, the reflexion techniques are not directly applicable, so we propose an alternative construction of an extension operator based on concatenation with gexponential functions.To do this, it will be necessary to extend the definition the g-exponential function in [5, Definition 4.5] to the whole real line.
Proof.We will assume, without loss of generality, that x y.We study different cases.
If y > x ≥ α, the results follows from the definition of the g-exponential.On the other hand, for x < y < α, given β < x, Finally, in the case x α y, we can proceed by combining the two previous cases.
We now prove the analogous result for the interval (−∞, α).
Proof.On the one hand, Thus, as in the previous case, it is enough to show that exp g (λ; α, •) ∈ L p g ((−∞, α)), for every p ∈ [1, ∞).We have again two cases.The case p = ∞ is obvious given that exp g (λ; α, t) we have that On the other hand, lim and, therefore, The case x α y can be obtained by combining the previous two cases.
In the following lemma we will prove that it is possible to concatenate two functions in W 1,p g under certain conditions.This will allow us to construct an extension operator.
Lemma 5.16.Let u 1 ∈ W 1,p g (I 1 ) y u 2 ∈ W 1,p g (I 2 ) be such that I 1 ∩ I 2 = {α} ⊂ and x < y, ∀x ∈ I 1 , ∀ y ∈ I 2 (observe that I = I 1 ∪ I 2 is an interval open from the right and closed from the left).Assume that u 1 (α) = u 2 (α) and define where u 1 and u 2 are given by the definition of W 1,p g (I 1 ) and W 1,p g (I 2 ) respectively.Then, u, u ∈ L p g (I) and In particular, u ∈ W 1,p g (I) and Proof.On the one hand, ) .The case of u is analogous.
On the other hand, given x, y ∈ I, we have that, whether x, y ∈ I 1 or x, y ∈ I 2 , it holds that Let us study the case x ∈ I 1 and y ∈ I 2 .Then, Now, we will use the previous results to show that, for a < b, it is possible to extend functions in In particular, we have the following result.
Theorem 5.17 (Extension operator).Let a < b and 1 p ∞. Then there exists a continuous linear operator P : where C > 0 is a constant depending only on p and µ g ([a, b)).
Proof.Given f ∈ W 1,p g ([a, b)), we define where λ − > 0 is any element of choice, for instance, we can take λ − = 1, and λ + > 0 given by Remark 5.14.Thanks to Lemmas 5.13 and 5.15, we have that P − f ∈ W 1,p g ((−∞, a)) and Let us consider P : By Lemma 5.16, we have that P f ∈ W 1,p g ( ).By this property and the definition of P, it is clear that property 1 in the statement of the theorem holds.Furthermore, and ).Therefore, we have proven properties 2 and 3.
f 0 , so property 4 holds as well.
Let us see now that W 1,1 g (I) is compactly embedded into L p g (I) for every 1 p < ∞. 1}.
To prove the thesis of the theorem, it is enough to show that is a relatively compact subset of For convenience, we may assume g c ( ) = and t∈D g ∆g(t) < ∞.Otherwise, we can always construct g : → such that g(t) = g(t), ∀t ∈ [a, b], g c ( ) = and t∈D g ∆ g(t) < ∞.
Let I = [a, b), P the extension operator given by Theorem 5.17 and = P( ).Let us show that is a relatively compact subset of L p g ( ).In order to achieve this goal, it is enough to check that the conditions of Theorem 4.12 hold.In order to simplify the notation, we will denote by f both the function defined in I itself and its extension as given by Theorem 5.17.Condition 1 in Theorem 4.12 states that { f (d n ) : f ∈ } ⊂ is bounded for each d n ∈ D g .Indeed, by Theorem 5.17, point 4, C, for every n ∈ and f ∈ where C is the embedding constant of W Therefore, given ǫ > 0, there exists n ∈ such that Condition 3 in Theorem 4.12 can be checked by observing that, given R > 0 and f where A R = {x ∈ : |x| > R} and γ is the pseudo-inverse of g c .Now, given R sufficiently large, In particular, taking into account the hypotheses on g c , lim from which it follows that for every ǫ > 0 there exists R > 0 such that , so 3 follows from (5.4).
Finally, we show that Theorem 4.12, point 4, holds.Given f ∈ and h > 0 (the case h < 0 is analogous), Now, on the one hand, If we integrate the first term on , For the second term we have that To summarize, Finally, where C does not depend on f .Taking ρ sufficiently small and |h| < ρ we show that Theorem 4.12, point 4, holds.

An application to decomposable functions
In this section we will apply the compactness results obtained in the previous sections to the space of those functions f : [a, b] → that can be expressed as the sum of a continuous function and a jump function.We characterize these functions in the following result.
) be such that f (t + ) exists for every t ∈ D g ∩ [a, b).The following are equivalent:

The function h(t)
Proof.Observe that we can talk of ∆ f at the points of D g because, by hypothesis, f (t + ) exists at the points of D g .The case where D g ∩ [a, b) is finite is straightforward, so we will deal only with the case where ) and, therefore, convergent.
g is well defined on the points of D g because f (t + ) exists at the points of D g .h is µ g -measurable because h = 0 in [a, b]\D g and D g is countable and g-measurable.In fact, so f B is well defined and, by the Fundamental Theorem of Calculus, Hence, by the Fundamental Theorem of Calculus, f B ∈ AC g B ([a, b], ).Also, since 3⇒4, we have that and we conclude that f B is bounded.
Since f and f B are g-continuous and bounded, so is Remark 6.2.The Riemann series theorem states that a convergent series is conditionally convergent if and only if the terms of the series can be rearranged in such a way that the sum of the new series is any fixed real number (or ±∞).As a consequence of this theorem, a convergent series is absolutely convergent if and only if the series of any subsequence of the sequence of terms is convergent.Points 1 and 2 in Lemma 6.1 are reminiscent of this result, so it is only natural to wonder whether we can prove a similar, but less restrictive, lemma where we do not deal with the absolute convergence of t∈D g |∆ f (t)| (remember that D g is countable, so this sum can be interpreted as a series in the traditional sense), but with some notion of conditional convergence of t∈D g ∆ f (t) or, even better, of s<t ∆ f (s) for every t ∈ [a, b] (see Lemma 6.1.4).This is not in general possible.The conditional convergence of series arises from the well order of the natural numbers and, although D g (or D g ∩ [a, t) for some t) inherits the total order of the real numbers, it is not in general a well order, so any definition of s<t ∆ f (s) would rely on a particular choice of a well order {t n } n∈ = D g and, for a different order, the sum may yield different results.Now we will study those cases where the decomposition can be taken as a product instead of a sum.This necessitates of the following result.Let ln : + → be the real logarithm, arg σ : \{0} → [σ − π, σ + π) be the σ-branch of the argument function (that is, z = |z| e i arg σ z ) and let log σ : \{0} → be the σ-branch of the logarithm (that is, log σ z = ln |z| + i arg σ z).We denote by log the principal branch of the complex logarithm, that is log 0 .Proposition 6.6.Let f ∈ BC g ([a, b], ) be such that for every t ∈ D g there exists f (t + ).Define D g, f = {t ∈ D g : ∆ f (t) = 0}.Assume that for every t ∈ D g, f there exists δ ∈ (0, b − t) such that f (s) = 0 for every s ∈ [t, t + δ).  ).Since γ (the pseudoinverse of g C ) is strictly increasing, γ −1 (D g ) is at most countable, so conditions 3 and 4 in Theorem 4.12 are satisfied (the functions involved are zero except on sets of null Lebesgue measure).Condition 2 translates as for every ǫ > 0, f ∈ S and some n ǫ , so applying Theorems 3.4 and 4.12 we obtain the result.

Definition 2 . 1 .
A function f : [a, b] → is g-continuous at a point t ∈ [a, b] if for every ǫ > 0 there exists δ > 0 such that | f (t) − f (s)| < ǫ, for all s ∈ [a, b], |g(t) − g(s)| < δ.g-continuity on [a, b] and uniform g-continuity are defined analogously to the usual case.We denote by C g ([a, b]; ) the set of g-continuous functions on [a, b], and BC g ([a, b], ) the set of bounded g-continuous functions on [a, b].

Definition 2 . 6 .
We define the Stieltjes derivative, or g-derivative, of f : [a, b] → at a point t ∈ [a, b] as

Proposition 2 .
10 ([3, Theorems 7.5 and 8.2, Exercise 15 p. 110]).Let (M , d) be a metric space, A ⊂ M .The following are equivalent: 1.A is totally bounded; 2. Every sequence in A has a Cauchy subsequence.Furthermore, if (M , d) is complete, the previous conditions are equivalent to 3. A is relatively compact.

Proof.
We start by making sure that the sets of the form (c, d] ∈ τ g (with d ∈ D g and c any real number) are the countable union of sets in .If c ∈ D g , we are done and, if c / ∈ D g , we necessarily have that (c, c + 1/n) ⊂ C g for any n (otherwise (c, d] / ∈ τ g ).Taking this into account, there exists a sequence {c n } n∈ ⊂ (c, d] ∩ ( ∪ D g ∪ N + g ) converging to c, and such that f (c n ) < f (t) for every t > c n .Under this conditions, (c n , d] ∈ , y (c, d] = n∈ (c n , d].A similar argument is valid for (a, b) ∈ τ g , with a, b ∈ .Any element of τ g is the union (possibly arbitrary) of open g-balls (that is, sets of the form {t ∈ [a, b] : |g(t) − g(t 0 )| < r} where t 0 ∈ [a, b] and r > 0), which are always of the form

Remark 3 . 15 .
The domain of f in the statement of Theorem 3.13 can be restricted to an interval [a, b].Just extend f constantly for values t < a and t > b and apply Theorem 3.13.Then f • g| [a,b] = f .Remark 3.16.We can present the Theorem 3.13 in terms of locally bounded locally uniformly g-continuous functions, that meaning that we can change the conditions in the statement to f | [a,b] ∈ BUC g ([a, b], ) for every interval [a, b] ⊂ and f ∈ C( , ).We just have to apply Remark 3.15 to ever increasing sequence of intervals [a n , b n ] to obtain extensions f n .Since the proof is constructive, the functions f n | [a n ,b n ] = f m for [a n , b n ] ⊂ [a m , b m ] and we can thus define f on all of satisfying the desired properties.

Definition 5 . 1 (
The Stieltjes-Sobolev space W 1,p g (I, )).Let I ⊂ be an interval which is closed from the left (in the case it is bounded from below) and open from the right, and p ∈ [1, ∞].We define
The embedding of W 1,1 g ([a, b)) into L p g ([a, b)) is compact.Proof.Fix p ∈ [1, ∞).Let us consider a, b ∈ such that a < b and define

4 .
for every ǫ > 0 there exists n such that for any f ∈ S,k>n | f (d k )| < ǫ.Proof.Total boundedness of S is equivalent to the total boundedness of bothS = { f : f ∈ S} ⊂ BC g ([a, b], ) and S = { f ′ g χ D g : f ∈ S} ⊂ L 1 ([a, b],

Remark 3.10. In
Corollary 3.8, since f ∈ BC g ([a, b],) is continuous at each point of [a, b]\D g , and left continuous in [a, b], f is regulated if and only if its right-hand limit exists for every t ∈ D g .

10 (Kolmogorov-Riesz compactness theorem).
. for every ǫ > 0 there is an n such that for any x in the set,k>n |x k | p < ǫ p .Suppose that 1 p < ∞.A subset of L 8, Theorems 4,5].Theorem 4.9.If 1 p < ∞, a subset of ℓ p is totally bounded if and only if 1. it is pointwise bounded, 2p ( ) is totally bounded if and only if 1. for every ǫ > 0 there exists R such that, for every f ∈ , |x|>R Let us consider a Cauchy sequence {u n } n∈ ⊂ W 1,p g (I).We have, thanks to the definition of the norm in W 1,p g (I) and Lemma 5.5, that {u n } n∈ is a Cauchy sequence in L p g (I) and { u n } n∈ is a Cauchy sequence in L p g (I).Moreover, due to Lemma 5.5, {u n } n∈ is a Cauchy sequence in BC g ([a, b]), for all a, b ∈ , a < b.