On Focal Borel Probability Measures

: The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures.


Introduction
The notions of depth and focality appear naturally in the optimal design of transcranial magnetic stimulation (TMS) coils. In such a context, these ideas refer to how deeply an electromagnetic field can be induced to a certain 3-dimensional body. In the excellent work of [1], the electric field penetration was quantified with the half-value depth, d 1/2 , focality with the tangential spread, S 1/2 , defined as the half-value volume (V 1/2 ) divided by the half-value depth: Formula (1) was implemented in [2] (Equation (4.1)) as part of a constraint in a singleoptimization problem that pretends to minimize the stored energy in the coil: where L ∈ R N×N is the inductance matrix (symmetric and positive definite), ψ ∈ R N , S 1/2 is the focality, and S 0 1/2 is the corresponding focality of the coil 0. The Euclidean metric and the Lebesgue measure are implicitly used in Formula (1). Those are the standard metric and measure employed in physics and engineering because, among other reasons, the Euclidean metric and the Lebesgue measure do not satisfy pathological properties such as vanishing on nonempty open subsets. However, in abstract topology and abstract measure theory, the existence of pathological metrics and measures is quite normal. Despite this, abstract measure theory has many applications not only in other areas of mathematics, but also in different disciplines such as physics or bioengineering. This manuscript takes the concept of focality as the motivating basis to add it to a more general and abstract scope. We introduce the novel concepts of focal continuous real-valued mappings and focal (regular) Borel probability measures, unveiling their geometric and topological properties. The novelty of this approach consists in the relationship with focal continuous real-valued functions and in establishing connections to regular Borel measures with finite variation. Among other results, we prove the existence of focal Borel probability measures on compact metric spaces (Theorem 3). We also demonstrate that the set of focal (regular) Borel probability measures is a convex but not extremal subset of the set of (regular) Borel probability measures (Theorem 5). In this way, we give an example that provides interesting information about the geometry of the unit ball of the dual of the space of real-valued continuous functions on K, C (K), where K is a compact Hausdorff topological space.

Preliminaries
If X is a topological space, then B(X), or simply B if there is no confusion with X, stands for the Borel σ-algebra of X, that is, the smallest σ-algebra of X containing the closed subsets of X. The elements of B(X) are called the Borel subsets of X. A Borel measure µ on X is a σ-additive measure defined on B(X) with values in a Hausdorff topological left-module M over a Hausdorff topological ring R, that is, a mapping µ: B(X) → R M satisfying that for every pairwise-disjoint sequence (A n ) n∈N of Borel subsets of X, µ( n∈N A n ) = ∑ n∈N µ(A n ). We focus on regular Borel probability measures on compact Hausdorff topological spaces. Explicit examples are given in Appendix B.
For a general metric space X, notations B X (x, r) and S X (x, r) stand for the closed ball of center x ∈ X and radius r > 0 and the sphere of center x ∈ X and radius r > 0, respectively. If X is a normed space, then B X and S X stand for the closed unit ball and the unit sphere, respectively.

Results
This section is divided into four subsections. In the first, a classical measure theory result on the measure of the union of increasing countable families of measurable subsets is extended to uncountable families. In the second, we define focality for real-valued continuous functions on a compact Hausdorff topological space. The third subsection focuses on the focality of (regular) Borel probability measures. Lastly, the fourth subsection shows that the set of focal (regular) Borel probability measures is a convex but not extremal subset of the set of (regular) Borel probability measures.

Increasing/Decreasing Families of Measurable Subsets
A classical measure theory result establishes that the measure of the union a countable increasing family of measurable subsets can be computed as the limit of the sequence of the measures of the subsets. This result was transported in [8] to the scope of measures defined on a effect algebra and valued on a topological module over a topological ring. Here, we extend [8] to uncountable families with countable cofinal subsets. However, we first recall [8] and prove it for the sake of completeness. Theorem 1. Let (Ω, Σ) be a measurable space. Let M be a Hausdorff topological module over a Hausdorff topological ring R. Let µ: Σ → M be a countably additive measure. If (A n ) n∈N ⊆ Σ is an increasing sequence of measurable subsets of Ω, then (µ(A n )) n∈N converges to µ( n∈N A n ).
If I is a directed set, and J ⊆ I is cofinal (see, for example, [9] (p. 461)), then any decreasing family of sets indexed by I satisfies that i∈I A i = j∈J A j . Indeed, it is clear that i∈I A i ⊆ j∈J A j and if a ∈ j∈J A j , then for every i ∈ I there exists j ∈ J with i ≤ j, so A j ⊆ A i , hence a ∈ A j ⊆ A i . Using the notion of cofinal set, we extend Corollary 1 to nets as follows.

Corollary 2. Let
(Ω, Σ) be a measurable space. Let M be a Hausdorff topological module over a Hausdorff topological ring R. Let µ: Σ → M be a countably additive measure. Let I be a nonempty directed set that has a countable cofinal subset J ⊆ I. If (A i ) i∈I is a decreasing family of measurable subsets of Ω such that i∈I A i is measurable, then the net (µ(A i )) i∈I converges to µ( i∈I A i ).
Proof. Suppose, on the other hand, that (µ(A i )) i∈I does not converge to µ( i∈I A i ). Then, we can find a neighborhood W of µ( i∈I A i ), such that, for all i ∈ I. there exists k ∈ I with k ≥ i, such that µ(A k ) / ∈ W. Let us write J = (j n ) n∈N . We construct an increasing sequence (k n ) n∈N on I using induction. For n = 1, we choose a k 1 ∈ I, such that k 1 ≥ j 1 and µ(A k 1 ) / ∈ W. Assume that, for some n ∈ N, we had already defined k 1 , . . . , k n , and take k n+1 ∈ I, such that Since j n ≤ k n for all n ∈ N, and J is cofinal in I, then (k n ) n∈N is cofinal in I. Therefore, However, the previous equality contradicts the fact that µ(A k n ) / ∈ W for every n ∈ N.
The final corollary of this first subsection displays the version of the previous result for increasing uncountable families with a countable cofinal subset. We spare the reader the details of the proof. Corollary 3. Let (Ω, Σ) be a measurable space. Let M be a Hausdorff topological module over a Hausdorff topological ring R. Let µ: Σ → M be a countably additive measure. Let I be a nonempty directed set that has a countable cofinal subset J ⊆ I. If (A i ) i∈I is an increasing family of measurable subsets of Ω such that i∈I A i is measurable, then net (µ(A i )) i∈I converges to µ( i∈I A i ).

Focality of Continuous Functions
We begin by defining the notion of focality for continuous real-valued functions with respect to a certain measure. However, we first need to introduce the regions of interest.
is usually called an α-region.
Obviously, the net of α- Definition 2 (Open α-region). Let K be a compact Hausdorff topological space. If f ∈ C (K) and As a consequence, if 0 ≤ α < 1, then every α-region has a nonempty interior because The next result shows that, if µ is a Borel probability measure on K, then µ(K 1 ( f )) can be obtained as the limit of the net (µ(K α ( f ))) α∈[0, 1) .
In many physics problems [2], α-regions that are of interest are those with a positive measure. This motivates the following definition.

Remark 1.
Let X be a metric space. Let A ⊆ X a nonempty subset of X. Function is nonexpansive.
In general, it is clear that not all real-valued nonexpansive mappings on a metric space have the form described in (9). Nevertheless, distance functions combined with translations allow for us to obtain a wide variety of properties. For example, every nonexpansive real function on a metric space is bounded by a distance function and a constant: Remark 2. Let X be a metric space and x 0 ∈ X. Then, every Lipschitz function f : Furthermore, in connection with the α-regions, we have the following result. If K is a compact metric space, then K is bounded, that is, it has finite diameter diam(K) := sup{d(k, l) : k, l ∈ K} < ∞. Proposition 2. Let K be a nonsingleton compact metric space. Let u ∈ K and α ∈ (0, 1). Function satisfies the following: 1. ψ u is positive and nonexpansive. 2. Proof. We only prove item 4. We spare the reader the details of the rest of the items. We have that Lastly, given u ∈ K and r > 0, taking α = 1 − min{r, diam(K)}/(2diam(K)), we obtain Then, every open subset of K is a union of sets O α (ψ u ).

Focality of Measures
There exist Borel probability measures on compact Hausdorff topological spaces that vanish at certain nonempty open subsets. For instance, if K is a nonsingleton compact Hausdorff topological space and k ∈ K, we can consider the regular Borel probability measure Since K is Hausdorff and not a singleton, U := K \ {k} is a nonempty open subset of K satisfying δ k (U) = 0.

Theorem 3.
If K is a compact metric space, then P f (K) ∩ rca(K) = ∅.
Proof. Let (k n ) n∈N ⊆ K be a dense sequence in K and define µ: = ∑ ∞ n=1 δ kn 2 n . µ ∈ C (K) * = rca(K) because C (K) * is a Banach space and ∑ ∞ n=1 δ kn 2 n is an absolutely convergent series in C (K) * (keep in mind that δ k n = 1 for all n ∈ N). We show that µ ∈ P f (K). Let U be a nonempty open subset of K. Since (k n ) n∈N is dense in K, there exists n 0 ∈ N such that k n 0 ∈ U. Then Lastly, Theorem 2 ensures that µ ∈ P f (K).
In compact metric spaces, in order to check whether a measure is focal, it is only necessary to look at the nonexpansive mappings.
Definition 6 (Weakly focal measure). Let K be a compact metric space. A Borel probability measure µ on K is weakly focal (w-focal) if every nonexpansive f ∈ C (K) is µ-focal. The set of weakly focal Borel probability measures on K are denoted by P wf (K).
We show that w-focal Borel probability measures coincide with focal probability measures. Proof. By definition, if µ is focal, then it is w-focal. Conversely, suppose that µ is weakly focal. We prove that µ(U) > 0 for every nonempty open subset U ⊆ K and then call on to Theorem 2. Indeed, fix an arbitrary nonempty open subset U ⊆ K. We may assume that U K since µ(K) = 1. Take f := d(•, K \ U). Since U is not empty, for every u ∈ U,

Extremal Structure of the Set of Focal Borel Probability Measures
The following result on this manuscript shows that P f (K) is a convex subset of P(K), but it is not extremal in P(K). In the next definition we recall the notion of extremal subset.

Definition 7 (Extremal subset).
A subset E of a subset D of a real vector space Z is extremal in D if E satisfies the extremal condition with respect to D: if d 1 , d 2 ∈ D and there exists α ∈ (0, 1) such that αd 1 + (1 − α)d 2 ∈ E, then d 1 , d 2 ∈ E.
We refer the reader to Appendix A for a further view on extremality theory and the geometry of normed spaces. Theorem 5. Let K be a nonsingleton compact Hausdorff topological space. If P f (K) = ∅, then P f (K) is a convex subset of P(K) but it is not extremal in P(K).
Since µ ∈ P f (K), we have that µ(U), µ(V) > 0, therefore µ(U), µ(V) < 1 and hence µ(K \ U), µ(K \ V) > 0. Consider the conditional probabilities of µ on U and K \ U, µ U and µ K\U , respectively, given by and Then, µ U , µ K\U ∈ P(K) \ P f (K) because µ K\U (U) = 0 and µ U (V) = 0. We demonstrate that 1 2 µ U + 1 2 µ K\U ∈ P f (K), reaching the conclusion that P f (K) is not extremal in P(K). Indeed, let W be any nonempty open subset of K. We have two options: In this case, W ⊆ K \ U, therefore because W is a nonempty open subset of K and µ ∈ P f (K). As a consequence, In the upcoming results, we reproduce Theorem 5 for regular measures to adapt it to rca(K). Given a topological space X, a countably additive measure µ: B(X) → [0, ∞] is inner regular provided that every Borel subset B of X is inner regular: µ(B) = sup{µ(F) : F ⊆ B, F compact}. µ is also an outer regular if every Borel subset B of X is outer regular: µ(B) = inf{µ(U) : U ⊇ B, U open}. Lastly, µ is regular if it is inner and outer regular. If B ∈ B(X) and µ(B) = 0, then B is trivially inner µ-regular, and if µ(B) = µ(X), then B is trivially outer µ-regular. If X is Hausdorff, and µ is finite and inner regular, then µ is outer regular. Conversely, if X is compact, and µ is finite and outer regular, then µ is inner regular. Then: 1.
If µ is inner regular, then so is µ A .

2.
If µ is outer regular and A is closed, then µ A is outer regular.

3.
If µ is finite and outer regular, then µ A is outer regular.
Fix an arbitrary B ∈ B(X). There exists a sequence (F n ) n∈N of compact subsets of X, such that F n ⊆ B ∩ A for every n ∈ N and (µ(F n )) n∈N converges to µ(B ∩ A). Since F n = F n ∩ A for all n ∈ N, we conclude that (µ A (F n )) n∈N converges to µ A (B). As a consequence, Fix an arbitrary B ∈ B(X). There exists a sequence (U n ) n∈N of open subsets of X such that B ∩ A ⊆ U n for every n ∈ N and (µ(U n )) n∈N converges to µ(B ∩ A). For every n ∈ N, V n := U n ∪ (X \ A) is open and satisfies that B ⊆ V n , B ∩ A ⊆ V n ∩ A = U n ∩ A ⊆ U n , and µ(B ∩ A) ≤ µ(V n ∩ A) = µ(U n ∩ A) ≤ µ(U n ). Therefore, (µ(V n ∩ A)) n∈N converges to µ(B ∩ A), meaning that (µ A (V n )) n∈N converges to µ A (B). As a consequence, µ A (B) = inf{µ A (U) : U ⊇ B, U open}.

3.
Let B ∈ B(X) and denote r = inf{µ(W ∩ A) : W ⊇ B, W open}. We prove that r ≤ µ(B ∩ A). Since µ is outer regular, we have Given an open subset W of X with B ⊆ W, since µ is finite, it holds that Therefore, However, we then arrive to the contradiction The following example displays a pathological measure for which there exists an outer regular Borel subset that is not inner regular for a conditional measure.
Finally, if U ⊆ X is open and contains B, then U = X since B is not open, thus

This way
is a convex subset of P(K) ∩ rca(K), which is itself a convex subset of S rca(K) , where S rca(K) denotes the unit sphere of rca(K). As usual, B rca(K) denotes the (closed) unit ball of rca(K).

Corollary 4.
Let K be a nonsingleton compact Hausdorff topological space. If P f (K) ∩ rca(K) = ∅, then P f (K) ∩ rca(K) is not a face of B rca(K) .
Consider the conditional probabilities of µ on F := cl(U) and G := K \ U, µ F and µ G . In view of Lemma 1, µ F , µ G ∈ P(K) ∩ rca(K). Thus, 1 2 µ F + 1 2 µ G ∈ P(K) ∩ rca(K). Since µ F (V) = 0 = µ G (U), we conclude that µ F , µ G / ∈ P f (K). Let us show that 1 2 µ F + 1 2 µ G ∈ P f (K), which finalizes the proof. Indeed, let W be any nonempty open subset of K. We have two options: • U ∩ W = ∅. Then because U ∩ W is a nonempty open subset of K and µ ∈ P f (K). • U ∩ W = ∅. In this case, W ⊆ K \ U = G, therefore because W is a nonempty open subset of K and µ ∈ P f (K).
Under the settings of Corollary 4, it is well known (see Appendix A and [12] (Theorem 3.7)) that P(K) ∩ rca(K) is, in fact, extremal in B rca(K) .

Discussion and Conclusions
If K is a nonsingleton compact Hausdorff topological space, then P f (K) is a convex subset of P(K) but not a face of P(K). However, as recalled in Appendix A, P(K) ∩ rca(K) is a face of B rca(K) , where B rca(K) denotes the unit ball of rca(K). So, we have the chain of inclusions P f (K) ∩ rca(K) ⊆ P(K) ∩ rca(K) ⊆ B rca(K) , where the first convex set is not a face of the second, whereas the second is a face of the third. This provides valuable information about the geometry of B rca(K) ≡ B C (K) * . It would be interesting to unveil other geometric or topological pathologies satisfied by the convex set of focal regular Borel probability measures.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.