Lusin-type theorems for Cheeger derivatives on metric measure spaces

A theorem of Lusin states that every Borel function on $R$ is equal almost everywhere to the derivative of a continuous function. This result was later generalized to $R^n$ in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincar\'e inequalities, which admit a form of differentiation by a famous theorem of Cheeger.


Introduction
A classical theorem of Lusin [17] states that for every Borel function f on R, there is a continuous function u on R that is differentiable almost everywhere with derivative equal to f .
In [1], Alberti gave a related result in higher dimensions.He proved the following theorem, in which | • | denotes Lebesgue measure and Du denotes the standard Euclidean derivative of u.Here C > 0 is a constant that depends only on k.
In other words, Alberti showed that it is possible to arbitrarily prescribe the gradient of a C 1 0 function u on Ω ⊂ R k off of a set of arbitrarily small measure, with quantitative control on all L p norms of Du.
Moonens and Pfeffer [18] applied Alberti's result to show a more direct analog of the Lusin theorem in higher dimensions: Theorem 1.2 ([18], Theorem 1.3).Let Ω ⊂ R k be an open set and let f : Ω → R k be measurable.Then for any ǫ > 0, there is an almost everywhere differentiable function u ∈ C(R k ) such that (a) u ∞ ≤ ǫ and {u = 0} ⊂ Ω, (b) Du = f almost everywhere in Ω, and (c) Df = 0 everywhere in R k \ Ω.
These "Lusin-type" results for derivatives in Euclidean space have applications to integral functionals on Sobolev spaces [1], to the construction of horizontal surfaces in the Heisenberg group ( [2,11]) and in the analysis of charges and normal currents [18].In addition, we remark briefly that the results of Alberti and Moonens-Pfeffer have been generalized to higher order derivatives on Euclidean space in the work of Francos [10] and Haj lasz-Mirra [11], though we do not pursue those lines here.
The purpose of this note is to extend the results of Alberti and Moonens-Pfeffer, in a suitable sense, to a class of metric measure spaces on which differentiation is defined.
In his seminal 1999 paper, Cheeger [6] defined (without using this name) the notion of a "measurable differentiable structure" for a metric measure space.Cheeger showed that a large class of spaces, the so-called PI spaces, possess such a structure.A differentiable structure endows a metric measure space with a notion of differentiation and a version of Rademacher's theorem: every Lipschitz function is differentiable almost everywhere with respect to the structure.
The class of PI spaces includes Euclidean spaces, all Carnot groups (such as the Heisenberg group), and a host of more exotic examples like those of [5], [16], and [7].
We prove the following two analogs of the results of Alberti and Moonens-Pfeffer for PI spaces.All the definitions are given in Section 2 below.
Theorem 1.3.Let (X, d, µ) be a PI space and let {(U j , φ j : X → R kj )} j∈J be a measurable differentiable structure on X.Then there are constants C, η > 0 with the following property: Let Ω ⊂ X be open with µ(Ω) < ∞ and let {f j : U j ∩ Ω → R kj } j∈J be a collection of Borel functions.Then for all ǫ > 0 there is an open set A ⊂ Ω and a Lipschitz function u ∈ C 0 (Ω) such that ) for all j ∈ J, The constants C, η > 0 depend only on the data of X.
As u is Lipschitz, the Poincaré inequality (see Lemma 2.4) will allow us to also control the global Lipschitz constant of u in Theorem 1.3, and conclude that if the right-hand side is finite.That the bounds (1.6) and (1.7) involve the the chart functions φ j is in some sense inevitable, as one can easily discover by looking at the measurable differentiable structure (R, φ(x) = 2x) on R. Note also that, unlike in the Euclidean setting of Theorem 1.1, the notion of C 1 regularity is not defined in PI spaces.Thus, the natural regularity for our constructed function u in Theorem 1.3 is Lipschitz.
Our second result is the analog in PI spaces of Theorem 1.2: Theorem 1.8.Let (X, d, µ) be a PI space.Let {(U j , φ j : U j → R kj )} be a measurable differentiable structure on X.Let Ω ⊂ X be open, let ǫ > 0, and let {f j : U j ∩ Ω → R kj } be a collection of Borel functions.Then there is a continuous function u on X that is differentiable almost everywhere and satisfies for each j ∈ J, and (1.11) Lip u = 0 everywhere in X \ Ω.
In particular, Theorems 1.3 and 1.8 allow one to prescribe the horizontal derivatives of functions on the Heisenberg group, or to prescribe the one-dimensional Cheeger derivatives of functions on the Laakso-type spaces of [16] and [7].

Definitions and Preliminaries
We will work with metric measure spaces (X, d, µ) such that (X, d) is complete and µ is a Borel regular measure.If the metric and measure are understood, we will denote such a space simply by X.An open ball in X with center x and radius r is denoted B(x, r).If B = B(x, r) is a ball in X and λ > 0, we write λB = B(x, λr).
We generally use C and C ′ to denote positive constants that depend only on the quantitative data associated to the space X (see below); their values may change throughout the paper.
If Ω ⊂ X is open, we let C c (Ω) denote the space of continuous functions with compact support in Ω.We also let C 0 (Ω) denote the completion of C c (Ω) in the supremum norm.Any function in C 0 (Ω) admits a natural extension by zero to a continuous function on all of X.
Recall that a real-valued function u on a metric space (X, d) is Lipschitz if there is a constant L ≥ 0 such that |u(x) − u(y)| ≤ Ld(x, y) for all x, y ∈ X.The infimum of all L ≥ 0 such that the above inequality holds is called the Lipschitz constant of u and is denoted LIP(u).
Given a real-valued (not necessarily Lipschitz) function u on X, we also define its pointwise upper Lipschitz constant at points x ∈ X by Lip u (x) = lim sup Two basic facts about Lip are easy to verify.First, for any two functions f and g, Second, if f and g are Lipschitz functions, then A non-trivial Borel regular measure µ on a metric space (X, d) is a doubling measure if there is a constant C > 0 such that µ(B(x, 2r) ≤ Cµ(B(x, r)) for every ball B(x, r) in X.The existence of a doubling measure µ on (X, d) implies that (X, d) is a doubling metric space, i.e. that every ball can be covered by at most N balls of half the radius, for some fixed constant N .In particular, a complete metric space with a doubling measure is proper : every closed, bounded subset is compact.Definition 2.3.A metric measure space (X, d, µ) is a PI space if (X, d) is complete, µ is a doubling measure on X and (X, d, µ) satisfies a "(1, q)-Poincaré inequality" for some 1 ≤ q < ∞: There is a constant C > 0 such that, for every compactly supported Lipschitz function f : X → R and every open ball B in X, (Here the notations ffl E gdµ and g E both denote the average value of the function g on the set E, i.e., 1 µ(E) ´E gdµ.)This definition can be found in [13]; it is equivalent to other versions of the Poincaré inequality in metric measure spaces, such as the original one of [12].If X is a PI space, then the collection of constants associated to the doubling property and Poincaré inequality on X are known as the data of X.
In addition to providing a differentiable structure (see below), the PI space property of X will supply two other key facts for us, summarized in the following proposition.Proposition 2.4.Let (X, d, µ) be a PI space.Then there is a constant C > 0, depending only on the data of X, such that the following two statements hold: (a) X is quasiconvex, meaning that any two points x, y ∈ X can be joined by a rectifiable path of length at most Cd(x, y).(b) For any bounded Lipschitz function u on X, Proof.The first statement can be found in Theorem 17.1 of [6].The second can be found (in greater generality than we need here) in [9], Theorem 4.7.
The following definition is essentially due to Cheeger, in Section 4 of [6].The form we state can be found in Definition 2.1.1 of [14] (see also [4]).The notation •, • denotes the standard inner product on Euclidean space of the appropriate dimension.Definition 2.5.Let (X, d, µ) be a metric measure space.Let {U j } j∈J be a collection of pairwise disjoint measurable sets covering X, let {k j } j∈J be a collection of non-negative integers, and let {φ j : X → R kj } j∈J be a collection of Lipschitz functions.
We say that the collection {(U j , φ j )} forms a measurable differentiable structure for X if the following holds: For every Lipschitz function u on X and every j ∈ J, there is a Borel measurable function Furthermore, the function d j u should be unique (up to sets of measure zero).
We call each pair (U j , φ j ) a chart for the differentiable structure on X.For more background on differentiable structures (also called "strong measurable differentiable structures" and "Lipschitz differentiability spaces") see [14,3].
Note that the defining property (2.6) for a measurable differentiable structure can be more succinctly rephrased as Lip u− d j u(x),φj (x) = 0.The link between PI spaces and measurable differentiable structures is given by the following theorem of Cheeger, one of the main results of [6].(See also [14,15,3] for alternate approaches.)Theorem 2.7 ([6], Theorem 4.38).Every PI space X supports a measurable differentiable structure and the dimensions k j of the charts U j are bounded by a uniform constant depending only on the constants associated to the doubling property and Poincaré inequality of X.
If X supports a measurable differentiable structure, then it generally supports many other equivalent ones.For example, the sets U j may be decomposed into measurable pieces or the functions φ j rescaled without altering the properties in Definition 2.5.At times, it will be helpful to assume certain extra properties of the charts. (2.11) The definition of a normalized chart is a minor modification of the notion of a "structured chart", due to Bate ([3], Definition 3.6).The following lemma, essentially due to Bate, says that a given chart structure on X can always be normalized by rescaling and chopping.Lemma 2.12.Let X be a PI space and let {(U j , φ j : X → R kj )} j∈J be a measurable differentiable structure on X.Then there exists a collection of sets {U j,k } j∈J,k∈Kj such that • each set U j,k is contained in U j and • {(U j,k , (LIP(φ j )) −1 φ j )} j∈J,k∈Kj is a normalized measurable differentiable structure on X.
Proof.By Lemma 3.4 of [3], we can decompose each chart U j into charts U j,k such that the measurable differentiable structure {(U j,k , φ j )} satisfies (2.11).
As a final step, we decompose each U j,k into closed sets, up to measure zero, while maintaining the same chart functions.
For technical reasons, it will be convenient in the proofs of Theorems 1.3 and 1.8 that the measurable differentiable structure is normalized.That this can be done without loss of generality is the content of the following simple lemma.Lemma 2.13.To prove Theorems 1.3 and 1.8, we can assume without loss of generality that the measurable differentiable structure {(U j , φ j )} is normalized.
Proof.Assume that we can prove Theorems 1.3 and 1.8 if the charts involved are normalized.
Suppose (U j , φ j ) is an arbitrary (not necessarily normalized) measurable differentiable structure on X.Let Ω ⊂ X be open with µ(Ω) < ∞, let {f j : U j ∩ Ω → R kj } j∈J be a collection of Borel functions, and let ǫ > 0.
By Lemma 2.12, there is a normalized measurable differentiable structure on X, where each U j,k is contained in U j .Let g j,k = (LIPφ j )f j .Apply Theorem 1.3 to the normalized measurable differentiable structure (2.14), with the functions g j,k and the same parameter ǫ.We immediately obtain an open set A ⊂ Ω and a Lipschitz function u ∈ C 0 (Ω) that satisfy all four requirements of Theorem 1.3.
A similar argument applies to reduce Theorem 1.8 to the normalized case.
The original arguments of [1] and [18] to prove Theorems 1.1 and 1.2 use the dyadic cube decomposition of Euclidean space.We will use the analogous decomposition in arbitrary doubling metric spaces provided by a result of Christ [8].
Proposition 2.15 ([8], Theorem 11).Let (X, d, µ) be a doubling metric measure space.Then there exist constants c ∈ (0, 1), η > 0, a 0 > 0, a 1 > 0, and C 1 > 0 such that for each k ∈ Z there is a collection open subsets of X with the following properties: For each k ∈ Z and i ∈ I k , and for each t > 0, We refer to the elements of any ∆ k as cubes.The next lemma is one of the primary differences between our proof of Theorem 1.3 and the proof of Theorem 1.1 from [1].It allows us to replace a single-scale argument in [1] by an argument that uses multiple scales simultaneously, which will allow us to deal with the presence of multiple charts.Lemma 2.16.Let (X, d, µ) be a complete doubling metric measure space.Suppose that µ(X \ ∪ j∈J U j ) = 0, where the sets U j are disjoint and measurable.Fix γ > 0 and positive numbers {δ j } j∈J .Then we can find a collection T of pairwise disjoint cubes in X (of possibly different scales) such that the following conditions hold: (i) µ(X \ T ∈T T ) = 0.
(ii) There is a map j : T → J such that and for each T ∈ T .
Proof.Let us call a cube T ∈ ∆ k "good for j" if it satisfies (2.17) and (2.18) with j(T ) = j, and let us call T "good" if it is good for some j ∈ J. Finally, let us call T "bad" if it is not good.Write ∆ g k for the sub-collection of ∆ k consisting of good cubes.
We then define our collection of cubes T to be In other words, our collection consists of all cubes that are the first good cube among all their ancestors of scales below 1.Note that any two distinct cubes in T are disjoint: if not, then one would contain the other, forcing the larger one to be bad.
For each cube T in this collection T , define j(T ) to be a choice of j ∈ J such that T is good for j.The collection T and the map j : T → J then automatically satisfy condition (ii) of the Lemma.
To verify (i), we show that almost every point x ∈ X is contained in one of the cubes T ∈ T .Let so that µ(Z) = 0 by Proposition 2.15 (i).Let x ∈ X \ Z be a point of µ-density of some U j0 .We claim that x ∈ T for some T ∈ T .Suppose, to the contrary, that x / ∈ T for any T ∈ T .Then x lies in an infinite nested sequence of bad cubes.But this is impossible: if an infinite nested sequence of cubes satisfied then eventually some Q i would be good for j 0 , and the first such good cube would be in T .
So ∪ T ∈T T contains almost every point in (∪ j∈J U j ) ∩ (X \ Z), which is almost every point of X.
The following lemma will ensure that we obtain a Lipschitz function in our construction.Recall the definition of quasiconvexity from Proposition 2.4.
Lemma 2.19.Let X be a complete and quasiconvex metric space and let u : X → R be a function on X. Suppose that there are pairwise disjoint open sets A i ⊂ X (i ∈ I), and a constant L ≥ 0 such that Then u is 2CL-Lipschitz on X, where C is the quasiconvexity constant of X.
Proof.Without loss of generality, we may assume that A i = X for each i ∈ I, otherwise the lemma is trivial.Note also that the assumption (2.20) improves immediately to LIP(u| Ai ) ≤ L for each i ∈ I.
Fix points x, y ∈ X.We will show that where C is the quasiconvexity constant of X.
Case 1: Suppose that, for some i, j ∈ I, we have x ∈ A i and y ∈ A j .In this case, we may also suppose that i = j, otherwise (2.22) follows from the assumption (2.20).Let t 0 = inf{t : γ(t) / ∈ A i } and t 1 = sup{t : γ(t) / ∈ A j }.By basic topology, γ(t 0 ) ∈ ∂A i ⊂ B and γ(t 1 ) ∈ ∂A j ⊂ B. Thus, we have Case 2: Suppose that x ∈ A i for some i ∈ I and that y ∈ B (or vice versa).We then have that u(y) = 0 and Case 3: Suppose that x ∈ B and y ∈ B. Then u(x) = u(y) = 0.
Note that Lemma 2.19 is false without assumption (2.21), as the Cantor staircase function shows.
The following lemma is due to Francos ([10], Lemma 2.3).Although Francos stated it only for subsets of R n , the proof works equally well in our setting.
Lemma 2.23.Let X be a PI space and let f be a Borel function from an open set Ω ⊂ X, with µ(Ω) < ∞, into some R N .Then, for any ǫ > 0, there is a compact set K ⊂ Ω and a continuous function

Proof of Theorem 1.3
Our main lemma is the analog of Lemma 7 of [1]: Lemma 3.1.Let (X, d, µ) be a PI space and let {(U j , φ j : X → R kj )} j∈J be a normalized measurable differentiable structure on X. Suppose that Ω ⊂ X is open with µ(Ω) < ∞ and Ω = X, and that {f j : Ω → R kj } is a uniformly bounded collection of continuous functions, i.e., that sup j∈J f j ∞ < ∞.Fix α, ǫ > 0.
Then there exists a compact set K ⊂ Ω and a Lipschitz function u ∈ C c (Ω) such that the following conditions hold: The constants η, C ′ > 0 depend only on the data of X.

and
(3.7) Using Lemma 2.16, we find a collection T of pairwise disjoint cubes covering almost all of X, and a map j : T → J such that
for each T ∈ T .Consider the sub-collection consisting of all cubes T ∈ T such that T ∩ K ′ = ∅.Index these cubes {T i } i∈I , and write j(i) for j(T i ).By (3.7) and (3.9), each cube T i (i ∈ I) lies in Ω.
For each i ∈ I, define S i ⊂ T i as where k is such that T ∈ ∆ k and t = (ǫ/4C 1 ) 1/η is fixed.This value of t was chosen to ensure (by Proposition 2.15 (iii)) that Note that S i is a compact subset of the open set T i .Let z i be a "center" of T i as in Proposition 2.15 (ii), so that T i both contains and is contained in a ball centered at z i of radius approximately diam T i .
For each cube T i in our collection, define a i ∈ R k j(i) by Note that the collection {|a i |} i∈I is bounded, because the collection {f j } j∈J is uniformly bounded.Let ψ i : X → R + be a Lipschitz function such that . (Here C is some constant depending only on the data of X.)By slightly widening the regions where ψ is constant, we can also easily arrange that Lip ψi = 0 everywhere in S i and in X \ T i .
Define u : X → R by A simple calculation shows that, for each i ∈ I, (Here we used the assumption that the measurable differentiable structure is normalized, and therefore LIP(φ j ) ≤ 1 for each j ∈ J.) Thus, as u = 0 outside i∈I T i , we see that u is Lipschitz on X by Lemma 2.19.In addition, u ∈ C c (Ω), with supp u ⊂ i∈I T i ⊂ Ω, and d j(i) u = a i a.e. on S i ∩ U j(i) .
Let K 1 = ∪ i∈I (S i ∩ U j(i) ), and let K be a compact subset of K 1 such that To verify (3.2), note that for each i ∈ I. Therefore, and so Let us now verify (3.3).Suppose that x ∈ U j ∩ K for some j ∈ J. Then x ∈ S i ∩ U j for some i ∈ I such that j(i) = j.Therefore, by (3.6) Finally, we must check (3.4) and (3.5).Observe that if T is a cube in T and x ∈ T , then the sum (3.10) defining u consists of at most one non-zero term.Therefore, for such x, we have by ( 2 Because almost every x ∈ X is contained in some T ∈ T , we have the bound (3.11) for almost every x ∈ Ω.
Recalling our normalization that LIP(φ j ) ≤ 1 for all j ∈ J, we see from (3.11) that, for all 1 ≤ p < ∞, Note that in the last inequality we used (3.8).
The case p = ∞, namely (3.5), follows from this by a limiting argument, or can alternatively be derived the same way.This completes the proof of Lemma 3.1.
Proof of Theorem 1.3.By Lemma 2.13, we may assume that the measurable differentiable structure is normalized.
It will also be convenient to assume that Ω is a proper subset of X, i.e., that Ω = X.We may assume this without loss of generality: If Ω = X, we replace Ω by Ω ′ = X \ {x 0 } for some arbitrary x 0 ∈ X. Proving Theorem 1.3 for Ω ′ also proves it for Ω.
Finally, we may also assume that ǫ < 1 and that sup j∈J f j ∞ > 0. We also extend each f j from U j ∩ Ω to all of Ω by setting f j = 0 off of U j .The proof now proceeds in two steps.
Step 1: Assume that the functions f j are uniformly bounded, i.e., that sup j∈J f j ∞ < ∞.
Let {α n } n≥1 be a decreasing sequence of positive real numbers with α 1 ≤ sup j∈J f j ∞ , to be chosen later.For each integer n ≥ 0, we will inductively build: a compact set K n ⊂ Ω, and a collection of continuous functions {f n j : Ω → R kj } j∈J .Let u 0 = 0.For each j ∈ J, we apply Lemma 2.23, to find a compact set K 0,j ⊂ Ω with µ(Ω \ K 0,j ) < 2 −1 2 −j ǫµ(Ω), and a continuous function f 0 j on Ω such that f 0 j = f j on K 0,j , (3.12) } j∈J have been constructed.Apply Lemma 3.1 to get a compact set Kn ⊂ Ω and a Lipschitz function for every j ∈ J and almost every x ∈ U j ∩ Kn , For each j ∈ J, apply Lemma 2.23 to find a compact set K n,j ⊂ Ω and a continuous map f n j = f n j on K n,j , and This completes stage n of the inductive construction.Now let Note that Purely for notational convenience, we now define a real-valued function for every p ∈ [1, ∞) and Note that, if p ∈ [1, ∞), we have, by (3.12) and (3.13), that In addition, F p is non-zero and finite for every p ∈ [1, ∞], by our assumption that 0 < sup j∈J f j ∞ < ∞.
We now calculate that, for p ∈ [1, ∞), almost everywhere in U j ∩ (Ω \ A), for each positive integer m.Thus, and both of these tend to zero as m tends to infinity.In the last inequality, we used the fact that (U j , φ j ) is a normalized chart, see Definition 2.8.Thus, f j = d j u a.e. on U j ∩ (Ω \ A), so (1.5) holds.This completes Step 1.
Step 2: The functions {f j : U j ∩ Ω → R kj } are arbitrary Borel functions.We first extend each f j to be zero off of U j , so that each f j is defined on all of Ω. Fix ǫ > 0. Choose r > 0 large so that B = {x : |f j (x)| > r for some j ∈ J} satisfies µ(B) < ǫ/2.Note that this is possible because, using the fact that f j = 0 off U j , we see that Then { fj } is a uniformly bounded collection of Borel functions on Ω such that, for all j ∈ J, | fj | ≤ |f j | everywhere and fj = f j outside the set B. Fix an open set A 1 ⊇ B such that µ(A 1 ) < ǫ/2.Then, for all j ∈ J, fj = f j outside of A 1 .Now apply the result of Step 1 to the uniformly bounded collection { fj }.We obtain an open set A 2 with µ(A 2 ) ≤ ǫ 2 µ(Ω) and a Lipschitz function u ∈ C 0 (Ω) such that Thus, for each j ∈ J, f j = d j u a.e. in U j ∩ (Ω \ A), where A = A 1 ∪ A 2 has µ(A) ≤ ǫµ(Ω).This verifies (1.4) and (1.5). If , which verifies (1.6).A similar calculation verifies (1.7).This completes the proof of Theorem 1.3.

4.
Proof of Theorem 1.8 In this section, we give the proof of Theorem 1.8.Given our Theorem 1.3, we can now just closely follow the proof given by Moonens-Pfeffer in [18].For the convenience of the reader, we give most of the details, although they are very similar to those of [18].
In our setting, the analog of Corollary 1.2 in [18] is the following: Lemma 4.1.Let X be a PI space with a normalized differentiable structure (U j , φ j : U j → R kj ).Let Ω ⊂ X be a bounded open subset of X and let {f j : U j ∩ Ω → R kj } be a collection of Borel functions.Then for every ǫ > 0, there exist a compact set K ⊂ U and a Lipschitz function u ∈ C c (Ω) such that for each j ∈ J, and Proof.We can again assume without loss of generality that Ω = X, otherwise we replace Ω = X by X \ {x 0 } for some x 0 ∈ X. Extend the functions f j to all of X by letting As in Step 2 in the proof of Theorem 1.3, we can find a compact set B ⊂ Ω ′ such that µ(Ω ′ \ B) < ǫ/4 and {f j } are uniformly bounded on B, i.e., sup j∈J f j L ∞ (B) = M < ∞.
For each j ∈ J, let g j = f j χ B , so the functions g j are uniformly bounded by the constant M > 0. Let η CM ), where C and η are the constants from Theorem 1.3.
Choose k large so that there are cubes Q (Note that the doubling property of µ and the boundedness of Ω implies that the collection {Q 1 , . . ., Q m } really is finite.) For each 1 ≤ i ≤ m, we now apply Theorem 1.3 to the collection {g j } in the cube Q i with parameter ǫ ′ = ǫ/8µ(Ω ′ ).For each 1 ≤ i ≤ m, we obtain a compact set K i ⊂ Q i with µ(Q i \ K i ) ≤ ǫ ′ µ(Q i ) and a Lipschitz function u i ∈ C c (Q i ) such that, for each j ∈ J, d j u i = g j almost everywhere in U j ∩ K i .
Furthermore, the remark after the statement of Theorem 1.3 shows that As u i ∈ C c (Q i ), it follows that, for each 1 ≤ i ≤ m, Let K = B ∩ (∪ m i=1 K i ), a compact subset of Ω.Our choices easily imply that µ(Ω \ K) < ǫ, which verifies (4.2).
Let u = m i=1 u i .Then u is a Lipschitz function in C c (Ω) that satisfies d j u = f j almost everywhere in U j ∩ K, so (4.3) holds.
Thus, the final condition (4.4) of Lemma 4.1 is verified.
We now prove Theorem 1.8.(To avoid some cumbersome subscripts, we change notation slightly and write Lip(g)(x) instead of Lip g (x).) Proof of Theorem 1.8.We again closely follow [18].
By Lemma 2.13, we may assume that the measurable differentiable structure is normalized.Without loss of generality, we also assume that ǫ < 1. Fix x 0 ∈ X and let B i = B(x 0 , i) for each i ∈ N.
We repeatedly apply Lemma 4.1.We inductively construct compact sets K i ⊂ Ω i = Ω ∩ B i \ ∪ i−1 k=1 K k and Lipschitz functions u i ∈ C c (Ω i ) such that, for each i ∈ N (4.5)Let K = ∪ ∞ i=1 K i and let u = ∞ i=1 u i .Note that u is a continuous function, because the bound u i ∞ ≤ 2 −i ǫ from (4.7) implies the uniform convergence of this sum.It also implies that u ∞ ≤ ǫ, verifying the first part of (1.9).
It remains to verify 1.10 and 1.11.We first claim that if x ∈ K i and k > i, then whenever x ∈ K i , y ∈ X, and k > i. Summing this over all k > i immediately proves (4.8).