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A study of measure-theoretic area formulas

We present a complete study of measure-theoretic area formulas in metric spaces, providing different measurability conditions.


Introduction
The problem of finding a proper notion of surface area has a long history in Calculus of Variations and in Geometric Measure Theory. Hausdorff measures, or more generally Carathéodory measures, are widely accepted as natural notions of surface area in metric spaces, [2, § 2.10.1]. On the other side, several important results of Geometric Measure Theory in Euclidean spaces are based on the fundamental concept of rectifiable set, whose Hausdorff measure can be computed by the classical area formula. Rectifiable sets can be considered in metric spaces, [2, §3.2.14], and in relation to their density properties, a general metric area formula was established in [7], see also [1] for a different approach.
Substantial obstructions appear when we wish to compute the Hausdorff measure (or the spherical measure) of purely unrectifiable sets. An important family of such sets naturally appears in the geometries of homogeneous groups, [3], where also smooth submanifolds can be purely unrectifiable. This is a consequence of their Hausdorff dimension, that can be strictly greater than their topological dimension, [5, § 0.6.B]. In this framework, the use of measure-theoretic area formulas seems to be the only approach to find integral formulas for the Hausdorff and spherical measure of submanifolds. In this connection, we mention some recent papers [4,6,9,11,12].
If the spherical measure is replaced by the centered Hausdorff measure, additional measuretheoretic area formulas have been proved in [4], leading to the relationship between perimeter measure and centered Hausdorff measure in stratified groups. In sum, applications to area formulas for submanifolds and to the perimeter measure in classes of homogeneous groups represent the first motivation of the present work. However, taking into account the deep relationship between density and rectifiability, [13,14], there are also interesting connections with the so-called Besicovitch 1 2 -problem and rectifiability, discussed in [8,10]. Considering that measure-theoretic area formulas are stated in general metric spaces, we would not be surprised about further applications.
We continue the study of measure-theoretic area formulas, according to which a suitable integration formula can be found between a Borel measure and a Carathéodory measure, [8]. The latter measure ψ ζ is obtained by a gauge ζ : S → [0, +∞] defined on a class of subsets S ⊂ P(X) of a metric space X. When ζ is proportional to some real power α > 0 of the diameter of a set, the Carathéodory measure coincides with the Hausdorff measure H α . In analogous way we obtain the spherical measure S α (Definition 2).
The general measure-theoretic area formula [8,Theorem 9] essentially arises from two key results: [2, Theorem 2.10.17 (2)] and [2, Theorem 2.10.18 (1)]. On the other hand, in considering a general ζ, it may happen that the quotients µ(S )/ζ(S ) in the upper covering limits are not well defined for some S ∈ S of possibly small diameter. The values of ζ(S ) may either vanish or be equal to +∞ and the same values may be taken by µ(S ). Another point in [2, Theorem 2.10.18 (1)] is that the gauge ζ need not be necessarily defined on the enlargementŜ of S , see (2.7).
Revised versions of [2, Theorem 2.10.17 (2)] and [2, Theorem 2.10.18 (1)] are stated in [8] without proof. Actually, all the results of [8] are only stated. The previous two theorems can be extended to our Lemmas 3.1 and 3.2, hence leading in turn to two more general measuretheoretic area formulas, proved in Theorems 4.1 and 4.2. All of these results provide a reasonably complete picture of measure-theoretic area formulas for Carathéodory measures in metric spaces, giving indeed the second motivation for this work.
To define the Federer density, we introduce the quotient function Q µ,ζ with respect to a measure µ over X and a gauge ζ, (Definition 3). We restrict Q µ,ζ to a subclass S µ,ζ ⊂ S where it is unambiguously defined. It is somehow surprising the fact that, despite this restriction, the Federer density F ζ (µ, x), defined in (2.5) by Q µ,ζ , still yields the following measure-theoretic area formula The cases where ψ ζ is either the Hausdorff measure or the spherical measure is important for applications, where the Federer density has also more chances to be explicitly found. This fact has been shown for submanifolds or intrinsic submanifolds in different classes of noncommutative homogeneous groups, [4,6,9,11,12].
The Federer density F ζ (µ, ·) is denoted by d α (µ, ·) for the Hausdorff measure and by s α (µ, ·) for the spherical measure. In these cases it is also important to show that d α (µ, ·) and s α (µ, ·) are measurable (or Borel), but this is not straightforward. We mention that when the metric space is a homogeneous group and a suitable restriction of ψ ζ is assumed to be locally doubling, the measurability of d α (µ, ·) and s α (µ, ·) is proved in [6,Proposition 2.3]. It is possible to show that in a metric space the Federer density d α (µ, ·) is Borel with respect to the subspace topology of a fixed set, according to Theorem 5.1. By a mild continuity assumption on the diameter function (Definition 4) and some more work, the previous Borel measurability can be extended to s α (µ, ·), see Theorem 5.6. Important consequences are the versions of (1.1) for the Hausdorff and spherical measure, established in Theorems 5.2, 5.3, 5.7 and 5.8.

Basic notions
Throughout the paper, X denotes a metric space equipped with a distance d. A covering relation is a subset C of {(x, S ) : x ∈ S ∈ P(X)}. Defining for A ⊂ X the class we say that C is fine at x, if for every ǫ > 0 there exists S ∈ C({x}) such that diam S < ǫ. Following the terminology of [2, §2.8.1], a nonempty family of sets F ⊂ P(X) covers A ⊂ X finely, if for each a ∈ A and ǫ > 0 there exists S ∈ F with a ∈ S ∈ F such that diam(S ) < ǫ. According to [2, § 2.8.16], the notion of covering relation yields the following notion of "upper" and "lower covering limits".
Definition 1 (Covering limits). If C is a covering relation which is fine at x ∈ X, and f : D → R, then we define the upper and lower covering limit, respectively as The closed ball and the open ball of center x ∈ X and radius r > 0 are denoted by respectively. The notion of δ-covering and the Carathéodory construction of [2, §2.10.1] are both recalled in the next definition.
Definition 2. If δ > 0, E ⊂ X and E = E j : j ∈ N ⊂ P(X) is a covering of E such that diam(E j ) ≤ δ for every j ∈ N, we say that E is a δ-covering of E. Let S ⊂ P(X) and let ζ : S → [0, +∞] be a fixed gauge. For R ⊂ X, we define The ζ-approximating measure, or Carathéodory measure, is defined as follows for every A ⊂ X. Denoting by F the family of closed sets of X, for α, c α > 0, we define Then the α-dimensional Hausdorff measure is H α = ψ ζ α . We define the set function as the restriction of ζ α to the family of all closed balls in X, that we denote by F b . Then ψ ζ b,α is the α-dimensional spherical Hausdorff measure, denoted by S α .
We will use the terminology "measure over X" to indicate an outer measure defined on all subsets of X, [2, § 2.1.2]. A gauge ζ : S → [0, +∞] is also fixed on a nonempty class S ⊂ P(X). The next definition gives the notion of Federer density, introduced in [8, Definition 4].
Definition 3 (Federer density). Let µ be a measure over X and let ζ : S → [0, +∞] be a gauge. We introduce the subfamily of sets along with the covering relation We choose x ∈ X and assume that C µ,ζ is fine at x. We define the quotient function Then the Federer density, or upper ζ-density of µ at x ∈ X, is well defined as We use a special notation when we consider Federer densities with respect to ζ α and ζ b,α , respectively. If µ is a measure over X and C µ,ζ b,α is fine at x ∈ X, then we set If C µ,ζ α is fine at x, then we set These densities will appear in the measure-theoretic area formulas for the Hausdorff and spherical measure, see Section 5.

Remark 1.
It is important to compute the upper limit (2.5) only over the sets S ∈ C µ,ζ , even if the definition of Q µ,ζ in (2.4) could be extended to any subset. Let us consider the gauge ζ α : F → [0, ∞], with ζ α (A) = c α diam(A) α for every A ∈ F , where α > 0. If we extend the definition of Q µ,ζ to all of F using (2.4), then for every x ∈ X it holds Indeed, for every ǫ > 0, we have The upper covering limit (2.6) now differs from the Federer density d(µ, x). It cannot give the measure-theoretic area formulas of Theorems 5.2 and 5.3.
In the sequel, a number τ > 1 is fixed and we will use the notion of enlargement of S , namely we introduce the set where S µ,ζ is defined in (2.3).

Preliminary tools
As mentioned in the introduction, the notion of Federer's density based on the quotient function (2.4) differs from the upper covering limits appearing in [2, Theorem 2.10.17 (2)] and [2, Theorem 2.10.18 (1)]. The next lemmas show that in these theorems the Federer density (2.5) can replace the upper covering limits.
We will use the notion of regular measure taken from [2, §2.1.5].
Lemma 3.1. Let ζ : S → [0, +∞] be a gauge and let A ⊂ X be nonempty. We consider a measure µ over X, t > 0 and assume that the following conditions hold.
Proof. Let B(µ, t, δ) be the set of all those points x ∈ E such that whenever S ∈ S µ,ζ and We claim that E = ∞ k=0 B(µ, t, 2 −k ). The hypothesis implies that for every fixed x ∈ E, there existsk ∈ N such that for every S ∈ S µ,ζ containing x and diam(S ) ≤ 2 −k , we have There are two possible cases: ζ(S ) = +∞ or In any case x ∈ B(µ, t, 2 −k ) and the claim is proved. We notice that

The last inequality holds in the case
where in the first equality we have used the assumption that µ is a regular measure.
An aspect to be clarified in [2, Theorem 2.10.18(1)] is that the gauge ζ in this theorem may not be necessarily defined on the enlargementŜ . In the next lemma, the conditions (3.1) take into account this fact and the upper covering limit in [2, Theorem 2.10.18(1)] is replaced by the Federer density.
Lemma 3.2. Let µ be a measure over X, B ⊂ X is nonempty and let ζ : S → [0, +∞] be a gauge. We assume that the following conditions hold.
(3) We have c ≥ 1 and η > 0 such that for every S ∈ S µ,ζ there existsS ∈ S such that whereŜ is the enlargement of S defined in (2.7).
Then for any open set V containing B, we have tψ ζ (B) ≤ µ(V).
The previous lemmas imply the following characterization of the absolutely continuity between two measures. Proposition 3.3. Let µ be a measure over X, let A ⊂ X be nonempty and consider a gauge ζ : S → [0, +∞]. We assume that the following conditions hold.
(4) A has a countable covering whose elements are open and have µ-finite measure.
(5) We have c ≥ 1 and η > 0 such that for every S ∈ S µ,ζ there existsS ∈ S such that whereŜ is the enlargement of S defined in (2.7). Then the condition µ A << ψ ζ A is equivalent to Proof. We first suppose that µ A << ψ ζ A. Let {V k } k∈N be a countable covering of A, whose elements are open and have µ-finite measure. By Lemma 3.2, for every t > 0 we have By our hypothesis, the equality (3.2) immediately follows. Conversely, we assume that (3.2) holds and consider B ⊂ A with ψ ζ (B) = 0. We set Finally, we can apply Lemma 3.1 to obtain that for all j ≥ 1, hence completing the proof.

Remark 2.
It is important to observe that in Proposition 3.3 the absolute continuity between the two measures follows from (3.2) without assuming condition (4).

Measure-theoretic area formulas
This section is devoted to the proof of the measure-theoretic area formula under two different groups of assumptions. Combining Lemma 3.1, Lemma 3.2 and Proposition 3.3, we get a first version of the measure-theoretic area formula. Let µ be a measure over X and consider a gauge ζ : S → [0, +∞]. We fix a nonempty set A ⊂ X and assume that the following conditions hold.
(1) µ is both a regular measure and a Borel measure.
A has a countable covering whose elements are open and have µ-finite measure. (6) We have c ≥ 1 and η > 0 such that for every S ∈ S µ,ζ there existsS ∈ S such that whereŜ is the enlargement of S defined in (2.7).
Letting h → 0 + , we conclude that µ(A 0 ) = 0. We choose a Borel set B ⊂ A and an arbitrary λ ∈ (0, 1), then we define These are all Borel sets. In particular, we have shown that replacing B with B +∞ (4.1) holds. Proposition 3.3 shows that µ({x ∈ A : F ζ (µ, x) = +∞}) = 0, therefore µ(B −∞ ) = 0. Since µ is Borel and assumption (5) We conclude that ψ ζ (B −∞ ) = 0 and (4.1) also holds with B −∞ in place of B. We are left to prove that (4.1) holds with B replaced by B ′ = B K (B +∞ ∪ B −∞ ). Using Lemma 3.1, for every ǫ > 0, we get Taking into account the arbitrary choice of ǫ and λ, we have proved that To prove the opposite inequality, we consider the nontrivial case where µ(B ′ ) < +∞. Taking into account that µ is Borel and the assumption (5) holds, for every ǫ > 0 and every k ∈ Z there exists an open set Using Lemma 3.2, we get We have proved that The arbitrary choice of ǫ and λ lead us to the opposite inequality hence concluding the proof. In the next version of the measure-theoretic area formula, the Borel measurability of the Federer density is replaced by the more general ψ ζ -measurability. We can also consider a more general ψ ζ -measurable domain of integration A, requiring in addition the σ-finiteness with respect to ψ ζ . Theorem 4.2 (Measure-theoretic area formula II). Let µ be a measure over X and consider a gauge ζ : S → [0, +∞]. We fix a nonempty set A ⊂ X and assume that the following conditions hold.
(2) Every element of S is Borel and every element of S µ,ζ is closed.
(4) A is ψ ζ -measurable and σ-finite with respect to ψ ζ . (5) A has a countable covering whose elements are open and have µ-finite measure. (6) We have c ≥ 1 and η > 0 such that for every S ∈ S µ,ζ there existsS ∈ S such that whereŜ is the enlargement of S defined in (2.7). (7) We have the absolute continuity µ A << ψ ζ A. If F ζ (µ, ·) : A → [0, +∞] is ψ ζ -measurable, then every ψ ζ -measurable subset B ⊂ A is also µ-measurable and the following formula holds Proof. Let B ⊂ A be a ψ ζ -measurable set. It is not restrictive to assume that ψ ζ (B) < +∞. This is justified by the fact that A can be written as a disjoint union i∈N A k , where A k is ψ ζ -measurable and it has ψ ζ -finite measure. Condition (2) implies that ψ ζ is Borel regular, so even if we may have a priori a sequence of arbitrary sets that cover A and have ψ ζ -finite measure, we can always assume they are Borel and in particular ψ ζ -measurable. Our claim for every B ∩ A k implies that this set is µ-measurable for every k ∈ N and The assumption that ψ ζ (B) < +∞ in particular gives Since ψ ζ is Borel regular and ψ ζ (B) < +∞, we can write B = B 0 ∪ N ⊂ A, where B 0 is Borel and ψ ζ (N) = 0. By our assumption, µ(N) = 0, hence B is also µ-measurable. Since A is σ-finite with respect to ψ ζ , actually any ψ ζ -measurable subset of A is also µ-measurable. We fix 0 < λ < 1 and for every k ∈ Z define that is ψ ζ -measurable, along with Thus, all of these sets are also µ-measurable. Since ψ ζ (B +∞ ) < +∞, for any fixed κ > 0, due to Lemma 3.1, we get µ(B +∞ ) ≤ κψ ζ (B +∞ ) < +∞.
The arbitrary choice of ǫ and λ, give the first inequality We now prove the opposite inequality, considering the nontrivial case where µ(B ′ ) < +∞. As before, the Borel regularity of µ and assumption (5) We obtain the following inequalities Our assumptions also allow to apply Lemma 3.2, hence It follows that The arbitrary choice of ǫ > 0 and λ provide the opposite inequality hence completing the proof.

Applications to Hausdorff measures
The special cases where ψ ζ is either the Hausdorff measure or the spherical measure are important for the applications in Geometric Measure Theory. The next theorems consider the case where the gauge is proportional to some power of the diameter. We prove that some of the "abstract conditions" for measure-theoretic area formulas are satisfied.
Proof. We have to prove that d α (µ, ·) is Borel with respect to the subspace topology of A. It suffices to prove that for any δ > 0 the function d α δ (µ, ·) : A → [0, +∞] defined as d α δ (µ, x) = sup{Q µ,ζ α (S ) : x ∈ S , S ∈ F µ,ζ α , diam(S ) < δ} is Borel with respect to the subspace topology of A. We fix t > 0 and consider the set We wish to prove that S t,δ is open in A. Let us fix y ∈ S t,δ . There exists a set S ∈ F µ,ζ α containing y, such that Q µ,ζ α (S ) > t and diam(S ) < δ.
Choosing a possibly smaller ǫ > 0, we also get where S ∪ B(y, ǫ) is still closed and belongs to F µ,ζ α . We have proved that Thus there exists an open set Ω such that S t,δ = Ω ∩ A and we may conclude that d α δ (µ, ·) is lower semicontinuous on A with respect to the subspace topology of A. Thus, y → d α (µ, y) is Borel with respect to the subspace topology of A.
Remark 5. We denote by F ′′ the family of closed bounded sets with positive diameter and observe that F ′′ = (F ′′ ) µ,ζ α . Then the assumption that F ′′ covers A finely implies that (F ) µ,ζ α covers A finely and from Theorem 5.1 we can conclude that d α (µ, ·) is well defined and Borel with respect to the subspace topology of A. If for every x ∈ A there exists a sequence of closed bounded sets F x,k (depending on x) that contain x and have positive diameter for every k ∈ N, with diam(F x,k ) → 0, then F ′′ covers A finely and so does (F ) µ,ζ α Theorem 5.2 (Area formula for the Hausdorff measure I). Let µ be a measure over X, α > 0 and fix A ⊂ X. We assume the validity of the following conditions.
(1) µ is both a regular measure and a Borel measure.
(2) F µ,ζ α covers A finely. Theorem 5.3 (Area formula for the Hausdorff measure II). Let µ be a measure over X, α > 0 and fix A ⊂ X. We assume the validity of the following conditions.
(3) A is H α -measurable and σ-finite with respect to H α . (4) A has a countable covering whose elements are open and have µ-finite measure. (5) We have the absolute continuity µ A << H α A.

Remark 6.
If F ′′ is as in Remark 5, then in the assumptions of either Theorem 5.2 or Theorem 5.3 the collection F ′′ covers A finely if and only if so does (F ) µ,ζ α . As noticed in Remark 5, from the equality F ′′ = (F ′′ ) µ,ζ α ⊂ (F ) µ,ζ α , if F ′′ covers A finely clearly so does (F ) µ,ζ α . Conversely, if (F ) µ,ζ α covers A finely, then we fix any x ∈ A, getting a sequence of closed sets F x,k ⊂ (F ) µ,ζ α that all contain x and such that diam(F x,k ) → 0. The absolute continuity of µ A with respect to S α A implies that (F ) µ,ζ α contains no points. As a result, for all k sufficiently large, we have F x,k ∈ F ′′ . From the arbitrary choice of x we infer that F ′′ covers A finely.
The next area formulas will replace the Hausdorff measure with the spherical measure. This requires a minimal regularity condition on the diameter function.
Definition 4 (Diametric regularity). A metric space X is called diametrically regular if for every x ∈ X there exist an R x > 0 and a δ x > 0 such that (0, δ x ) ∋ r → diam (B(y, r)) is a continuous function for every y ∈ B(x, R x ).
Lemma 5.4. If a metric space X is diametrically regular, then for every x ∈ X there exist R x > 0 and δ x > 0 such that whenever 0 < t < δ x and y ∈ B(x, R x ), we have diam(B(y, t)) = diam (B(y, t)).
Theorem 5.6. Let X be diametrically regular, α > 0 and let µ be a measure over X. If A ⊂ X and (F b ) µ,ζ b,α covers A finely, then s α (µ, ·) : A → [0, +∞] is Borel with respect to the subspace topology of A.
Proof. We consider the gauge ζ o,α of Lemma 5.5. The lemma shows that in our assumptions also (F o ) µ,ζ o,α covers A finely, so also the Federer density F ζ o,α (µ, x) is well defined at each point x ∈ A. Our first claim is the following equality for every x ∈ A. We fix x ∈ A and consider R x > 0 and δ x > 0 of Definition 4. We wish to prove that It is equivalent to showing that We fix δ > 0 such that δ < min{R x , δ x } and choose B ∈ (F o ) µ,ζ o,α such that x ∈ B and diam(B) < δ. We may write B = B(y, r) for some y ∈ X and r > 0. We define t y = sup{d(y, z) : z ∈ B} ≤ r and notice that clearly B ⊂ B(y, t y ), hence We observe that t y ≤ r and In particular, y ∈ B(x, R x ) and t y < δ x . Since B ⊂ B(y, t y ) ⊂ B(y, t ′ y ) for every t ′ y > t y , then diam(B) ≤ diam(B(y, t ′ y )) and by diametric regularity for t ′ y ց t y and t ′ y < δ x , we obtain (5.6) diam(B) ≤ diam(B(y, t y )).
To prove the opposite inequality, we first consider the case s α (µ, x) = +∞. Then we can find a family of closed balls {C k } k∈N ⊂ (F b ) µ,ζ b,α such that, for every k ∈ N we have diam(C k ) < 2 −k , x ∈ C k and lim k→+∞ Q µ,ζ b,α (C k ) = +∞.
If there exists k 0 ∈ N such that for every k ∈ N with k ≥ k 0 we have diam(C k ) > 0, then there existsr k > 0 such that C k = B(x k ,r k ) and the following supremum satisfies r k = sup {d(z, x k ) : z ∈ C k } > 0 and r k ≤r k .
We have seen that B(x k , r k ) ⊂ C k . On the other hand, by definition of r k we also get hence C k = B(x k , r k ). We select any r ′ k ∈ (r k , 2r k ] and observe that , where the last inequality follows from the definition of r k . It follows that as k → +∞. This esablishes the opposite inequality when the diameters of C k are positive for all large k's. In the remaining case there exists a strictly increasing sequence of integers α(k) ∈ N such that such that diam(C α(k) ) = 0 for all k ∈ N. It follows that C α(k) = {x} and µ({x}) > 0. We choose any deacreasing sequence s k ց 0, hence We have shown that F ζ o,α (µ, x) = +∞, hence concluding the proof of the inequality in the case s α (µ, x) = +∞. We will prove (5.9) when s α (µ, x) < +∞. It is convenient to restrict our attention to the following subset of integers where R x and δ x are those of Definition 4. By our assumptions, there exists a family of closed balls such that x ∈ D k for all k ∈ N and there exists k 1 ∈ N such that for every k ∈ Λ with k ≥ k 1 we have diam(D k ) < 2 −k and As a consequence, for these k's we have that ζ b,α (D k ) > 0 and We choose any k ∈ Λ with k ≥ k 0 . Arguing as before, we find a center y k ∈ D k and the following positive radius such that D k = B(y k , t k ). We know that d(y k , x) ≤ diam(D k ) < 2 −k < R x , therefore the diametric regularity and Lemma 5.4 imply that s → diam(B(y k , s)) = diam(B(y k , s)) is continuous on (0, δ x ). Then we can find t ′ k ∈ (t k , δ x ) such that 0 < diam(B(y k , t ′ k )) < 2 −k and .
We have proved that for all k ∈ Λ with k ≥ k 1 . As k → +∞, the opposite (5.9) follows and the claim ( when δ > 0 is arbitrarily chosen. We fix t > 0 and consider the set so that z ∈ S t,δ , hence B ∩ A ⊂ S t,δ . This shows that S t,δ is open in the subspace topology of A and in particular it is Borel with respect to this topology. Remark 7. If we denote by F ′ b the family of closed balls with positive diameter, then clearly F ′ b = (F ′ b ) µ,ζ b,α . As a result, if F ′ b covers A finely, then (F b ) µ,ζ b,α also covers A finely and the Federer density s α (µ, ·) of Theorem 5.6 is well defined. For instance, if for every x ∈ A there exists a sequence of closed metric balls B x,k (depending on x) with positive diameter and containing x, such that diam(B x,k ) → 0, then F ′ b covers A finely and so does (F ) µ,ζ b,α . Theorem 5.7 (Area formula for the spherical measure I). Let µ be a measure over X, α > 0 and fix A ⊂ X. We assume the validity of the following conditions.
(1) X is a diametrically regular metric space.
(2) µ is both a regular measure and a Borel measure.
(3) (F b ) µ,ζ b,α covers A finely. (4) A is a Borel set. Proof. Due to Theorem 5.6, the density s α (µ, ·) : A → [0, +∞] is Borel with respect to the subspace topology of A, hence by our assumptions it is also Borel. If D 0 = B(x, r) is any closed ball, we consider its enlargement Theorem 5.8 (Area formula for the spherical measure II). Let µ be a measure over X, α > 0 and fix A ⊂ X. We assume the validity of the following conditions.
(1) X is a diametrically regular metric space.
(4) A is S α -measurable and σ-finite with respect to S α . (5) A has a countable covering whose elements are open and have µ-finite measure. (6) We have the absolute continuity µ A << S α A. Then s α (µ, ·) : A → [0, +∞] is S α -measurable, every S α -measurable set B ⊂ A is also µ-measurable and we have Proof. In view of Theorem 5.6, the density s α (µ, ·) : A → [0, +∞] is Borel with respect to the subspace topology of A. Being A an S α -measurable set, then s α (µ, ·) is S α -measurable. Arguing as in the proof of Theorem 5.7, we observe that condition (6) of Theorem 4.2 is satisfied, again using the notion of enlargement of (2.7). We can apply Theorem 4.2, that concludes the proof.
Remark 8. Let F ′ b be the family of balls introduced in Remark 7. Then in the assumptions of either Theorem 5.7 or Theorem 5.8 we actually have (F b ) µ,ζ b,α = F ′ b . As already pointed out in Remark 7, it holds Conversely, any set of (F b ) µ,ζ b,α must differ from a point, due to the absolute continuity of µ A with respect to S α A, hence the opposite inclusion holds. Thus, in both Theorems 5.7 and 5.8 the assumption (3) can be replaced by the condition that F ′ b covers A finely.