On the role of the point at infinity in Deny’s principle of positivity of mass
for Riesz potentials

Natalia Zorii

Abstract. First introduced by J. Deny, the classical principle of positivity
of mass says that if $\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$
everywhere on $\mathbb{R}^{n}$, then
$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$. Here $\mu,\nu$ are positive
Radon measures on $\mathbb{R}^{n}$, $n\geqslant 2$, and $\kappa_{\alpha}\mu$
is the potential of $\mu$ with respect to the Riesz kernel $|x-y|^{\alpha-n}$
of order $\alpha\in(0,2]$, $\alpha<n$. We strengthen Deny’s principle by
showing that $\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$ still holds
even if $\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$ is fulfilled only on a
proper subset $A$ of $\mathbb{R}^{n}$ that is not inner $\alpha$-thin at
infinity; and moreover, this condition on $A$ can not in general be improved.
Hence, if $\xi$ is a signed measure on $\mathbb{R}^{n}$ with $\int 1\,d\xi>0$,
then $\kappa_{\alpha}\xi>0$ everywhere on $\mathbb{R}^{n}$, except for a
subset which is inner $\alpha$-thin at infinity. The analysis performed is
based on the author’s recent theories of inner Riesz balayage and inner Riesz
equilibrium measures (Potential Anal., 2021), the inner equilibrium measure
being understood in an extended sense where both the energy and the total mass
may be infinite.  000 2010 Mathematics Subject Classification: Primary 31C15.
000 Key words: Principle of positivity of mass for $\alpha$-Riesz potentials,
inner $\alpha$-thinness at infinity, inner $\alpha$-Riesz balayage, a
generalized concept of inner $\alpha$-Riesz equilibrium measure.

##  1\. A strengthened version of Deny’s principle of positivity of mass

In this paper we shall deal with the theory of potentials on $\mathbb{R}^{n}$,
$n\geqslant 2$, with respect to the $\alpha$-Riesz kernel
$\kappa_{\alpha}(x,y):=|x-y|^{\alpha-n}$ of order $\alpha\in(0,2]$,
$\alpha<n$, where $|x-y|$ is the Euclidean distance between
$x,y\in\mathbb{R}^{n}$. Denote by $\mathfrak{M}^{+}$ the cone of all positive
Radon measures $\mu$ on $\mathbb{R}^{n}$ such that the $\alpha$-Riesz
potential

$\kappa_{\alpha}\mu(x):=\int\kappa_{\alpha}(x,y)\,d\mu(y)$

is not identically infinite, which according to [19, Section I.3.7] occurs if
and only if

$\int_{|y|>1}\frac{d\mu(y)}{|y|^{n-\alpha}}<\infty.$ (1.1)

Note that the measures in question may be unbounded, namely with
$\mu(\mathbb{R}^{n})=+\infty$.

The principle of positivity of mass was first introduced by J. Deny (see e.g.
[12]), and for $\alpha$-Riesz potentials it reads as follows [16, Theorem
3.11].

######  Theorem 1.1.

For any $\mu,\nu\in\mathfrak{M}^{+}$ such that

$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu\text{ \ everywhere on
$\mathbb{R}^{n}$},$ (1.2)

we have $\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$.

It is easy to verify that (1.2) can be slightly weakened by replacing
‘everywhere on $\mathbb{R}^{n}$’ by ‘nearly everywhere on $\mathbb{R}^{n}$’
(see e.g. the author’s recent result [28, Theorem 2.6], establishing the
principle of positivity of mass for potentials with respect to rather general
function kernels on locally compact spaces). Recall that a proposition
involving a variable point $x\in\mathbb{R}^{n}$ is said to hold nearly
everywhere (n.e.) on $A\subset\mathbb{R}^{n}$ if $c_{\alpha}(E)=0$, where $E$
is the set of $x\in A$ for which the proposition fails to hold, while
$c_{\alpha}(E)$ is the inner $\alpha$-Riesz capacity of $E$ [19, Section
II.2.6] (cf. Sect. 2 below).

In the present study we shall show that Theorem 1.1 still holds even if the
assumption $\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$ is fulfilled only
on a proper subset $A$ of $\mathbb{R}^{n}$, which however must be ‘large
enough’ in an arbitrarily small neighborhood of $\infty_{\mathbb{R}^{n}}$, the
Alexandroff point of $\mathbb{R}^{n}$. This discovery illustrates a special
role of the point at infinity in Riesz potential theory, in particular in the
principle of positivity of mass.

To be exact, the following theorem holds true.

######  Theorem 1.2.

Given $\mu,\nu\in\mathfrak{M}^{+}$, assume there exists
$A\subset\mathbb{R}^{n}$ which is not inner $\alpha$-thin at infinity, and
such that

$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu\text{ \ n.e.\ on $A$}.$

Then

$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n}).$

######  Remark 1.3.

The concept of inner $\alpha$-thinness of a set at infinity was introduced in
[25, Definition 2.1]. Referring to Sect. 4 for details, at this point we only
note that $A\subset\mathbb{R}^{n}$ is not inner $\alpha$-thin at infinity if
and only if $y=0$ is inner $\alpha$-regular for $A^{*}$, the inverse of $A$
with respect to $\\{|x|=1\\}$. Such $A$ may in particular be thought of as
$\\{x_{i}>q\\}$, where $q\in\mathbb{R}$ and $x_{i}$ is the $i$-coordinate of
$x\in\mathbb{R}^{n}$, or as $Q^{c}:=\mathbb{R}^{n}\setminus Q$,
$Q\subset\mathbb{R}^{n}$ being bounded (see also Example 1.9). If $A$ is not
inner $\alpha$-thin at infinity, it is ‘rather large’ at infinity in the sense
that then necessarily $c_{\alpha}(A)=\infty$, whereas the converse is in
general not true (see Corollary 4.6, cf. Example 4.7 for illustration).

The following theorem shows that Theorem 1.2 is sharp in the sense that the
requirement on $A$ of not being $\alpha$-thin at infinity can not in general
be weakened.

######  Theorem 1.4.

If a set $A\subset\mathbb{R}^{n}$ is inner $\alpha$-thin at infinity, there
exist $\mu_{0},\nu_{0}\in\mathfrak{M}^{+}$ such that
$\kappa_{\alpha}\mu_{0}=\kappa_{\alpha}\nu_{0}$ n.e. on $A$, but nonetheless,
$\mu_{0}(\mathbb{R}^{n})>\nu_{0}(\mathbb{R}^{n})$.

Nevertheless, Theorem 1.2 remains valid for arbitrary $A\subset\mathbb{R}^{n}$
once we impose upon the measures $\mu$ and $\nu$ suitable additional
requirements (see Theorem 1.5).

A measure $\mu\in\mathfrak{M}^{+}$ is said to be concentrated on
$A\subset\mathbb{R}^{n}$ if $A^{c}$ is $\mu$-negligible, or equivalently if
$A$ is $\mu$-measurable and $\mu=\mu|_{A}$, $\mu|_{A}$ being the restriction
of $\mu$ to $A$. We denote by $\mathfrak{M}^{+}_{A}$ the cone of all
$\mu\in\mathfrak{M}^{+}$ concentrated on $A$. (For a closed set $A$, a measure
$\mu$ belongs to $\mathfrak{M}^{+}_{A}$ if and only if its support $S(\mu)$ is
contained in $A$.)

A measure $\mu\in\mathfrak{M}^{+}$ is said to be $c_{\alpha}$-absolutely
continuous if $\mu(K)=0$ for every compact set $K\subset\mathbb{R}^{n}$ with
$c_{\alpha}(K)=0$. This certainly occurs if $\int\kappa_{\alpha}\mu\,d\mu$ is
finite or, more generally, if $\kappa_{\alpha}\mu$ is locally bounded (but not
conversely, see [19, pp. 134–135]).

######  Theorem 1.5.

For any set $A\subset\mathbb{R}^{n}$ and any $c_{\alpha}$-absolutely
continuous measures $\mu,\nu\in\mathfrak{M}^{+}_{A}$ such that
$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$ n.e. on $A$, we still have
$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$.

######  Remark 1.6.

If $A\cap A_{I}=\varnothing$, where $A_{I}$ consists of all inner
$\alpha$-irregular points for $A$, the condition of the $c_{\alpha}$-absolute
continuity imposed on $\mu$ and $\nu$, is unnecessary for the validity of
Theorem 1.5. Namely, for any $A\subset\mathbb{R}^{n}$ such that $A\cap
A_{I}=\varnothing$, and any $\mu,\nu\in\mathfrak{M}^{+}_{A}$ with the property
$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$ n.e. on $A$, we still have
$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$. (See Sect. 3 for the
concept of inner $\alpha$-irregular point and relevant results.)

Let $\mathfrak{M}$ stand for the linear space of all real-valued (signed)
Radon measures $\xi$ on $\mathbb{R}^{n}$ such that (1.1) is fulfilled with
$\mu$ replaced by $|\xi|:=\xi^{+}+\xi^{-}$, where $\xi^{+}$ and $\xi^{-}$
denote the positive and negative parts of $\xi$ in the Hahn–Jordan
decomposition. Then the potential $\kappa_{\alpha}\xi$ of any
$\xi\in\mathfrak{M}$ is well defined and finite n.e. on $\mathbb{R}^{n}$
(actually, even everywhere on $\mathbb{R}^{n}$ except for a polar set, see
[19, Section III.1.1]).

The next two corollaries follow directly from Theorem 1.2.

######  Corollary 1.7.

For any $\xi\in\mathfrak{M}$ with
$\xi^{+}(\mathbb{R}^{n})>\xi^{-}(\mathbb{R}^{n})$, the potential
$\kappa_{\alpha}\xi$ is strictly positive everywhere on $\mathbb{R}^{n}$,
except for a subset that is inner $\alpha$-thin at infinity.

######  Corollary 1.8.

For any $\xi\in\mathfrak{M}$ with
$\xi^{+}(\mathbb{R}^{n})\neq\xi^{-}(\mathbb{R}^{n})$, the set of all
$x\in\mathbb{R}^{n}$ for which $\kappa_{\alpha}\xi(x)=0$ is inner
$\alpha$-thin at infinity.

######  Example 1.9.

Let $n=3$ and $\alpha=2$. Consider the rotation bodies

$F_{j}:=\bigl{\\{}x\in\mathbb{R}^{3}:\ 0\leqslant x_{1}<\infty,\
x_{2}^{2}+x_{3}^{2}\leqslant\varrho_{j}^{2}(x_{1})\bigr{\\}},\ j=1,2,$ (1.3)

where

$\displaystyle\varrho_{1}(x_{1})$ $\displaystyle:=x_{1}^{-s}\text{ \ with\
}s\in[0,\infty),$ $\displaystyle\varrho_{2}(x_{1})$
$\displaystyle:=\exp(-x_{1}^{s})\text{ \ with\ }s\in(0,\infty).$ (1.4)

By the strengthened principle of positivity of mass (Theorem 1.2), for any
$\mu,\nu\in\mathfrak{M}^{+}$ such that
$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$ n.e. on $F_{1}$, we have
$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$, the set $F_{1}$ being not
$2$-thin at infinity [25, Example 2.1]. On the other hand, the set $F_{2}$ is
$2$-thin at infinity [25, Example 2.1], and hence, according to Theorem 1.4,
there are $\mu_{0},\nu_{0}\in\mathfrak{M}^{+}$ such that

$\nu_{0}(\mathbb{R}^{n})<\mu_{0}(\mathbb{R}^{n}),\text{ \ though \
}\kappa_{\alpha}\mu_{0}=\kappa_{\alpha}\nu_{0}\text{\ n.e.\ on $F_{2}$}.$

(Actually, any nonzero positive measure concentrated on $F_{2}^{c}$ may serve
as $\mu_{0}$, and the Newtonian balayage of such a $\mu_{0}$ onto $F_{2}$ then
serves as $\nu_{0}$, see [24, Example 8.8].)

Figure 1. The set $F_{2}$ in Example 1.9 with
$\varrho_{2}(x_{1})=\exp(-x_{1})$.

The proofs of Theorems 1.2, 1.4, and 1.5 (see Sect. 5) are based on the theory
of inner $\alpha$-Riesz balayage as well as on the theory of inner
$\alpha$-Riesz equilibrium measures, both originated in [24, 25] (see also
[26]–[28] for some further relevant results). The inner $\alpha$-Riesz
equilibrium measure is understood in an extended sense where its energy as
well as its total mass may be infinite. To make the present study self-
contained, we give a brief summary of [24]–[28] (see Sects. 3, 4). To begin
with, we first review some basic facts of the theory of $\alpha$-Riesz
potentials, see [19].

##  2\. Basic facts of the theory of $\alpha$-Riesz potentials

In what follows we shall use the notations and conventions introduced in Sect.
1.

Throughout the paper the linear space $\mathfrak{M}$ is meant to be equipped
with the (Hausdorff) vague topology of pointwise convergence on the class
$C_{0}(\mathbb{R}^{n})$ of all continuous functions
$f:\mathbb{R}^{n}\to\mathbb{R}$ of compact support.

The Riesz composition identity
$\kappa_{\alpha}=\kappa_{\alpha/2}\ast\kappa_{\alpha/2}$ [20, Section 1, Eq.
(1)] (cf. also [19, Eq. (1.1.12)]) implies that the kernel $\kappa_{\alpha}$
is strictly positive definite, which means that for any (signed)
$\mu\in\mathfrak{M}$, the $\alpha$-Riesz energy

$\kappa_{\alpha}(\mu,\mu):=\int\kappa_{\alpha}(x,y)\,d(\mu\otimes\mu)(x,y)$

is ${}\geqslant 0$ whenever defined, and moreover it is zero only for zero
measure. This in turn implies (see e.g. [15, Lemma 3.1.2]) that all the
measures in $\mathfrak{M}$ of finite $\alpha$-Riesz energy form a pre-Hilbert
space $\mathcal{E}_{\alpha}$ with the inner product

$\langle\mu,\nu\rangle:=\kappa_{\alpha}(\mu,\nu):=\int\kappa_{\alpha}(x,y)\,d(\mu\otimes\nu)(x,y)$

and the energy norm $\|\mu\|:=\sqrt{\kappa_{\alpha}(\mu,\mu)}$. The
(Hausdorff) topology on $\mathcal{E}_{\alpha}$ defined by means of this norm
is said to be strong.

By Deny [11] (for $\alpha=2$, see also H. Cartan [9]), the cone
$\mathcal{E}^{+}_{\alpha}:=\\{\mu\in\mathcal{E}_{\alpha}:\ \mu\geqslant 0\\}$
is strongly complete, and the strong topology on $\mathcal{E}^{+}_{\alpha}$ is
finer than the (induced) vague topology.111In B. Fuglede’s terminology [15],
the Riesz kernel is, therefore, perfect. Thus every strong Cauchy net
$(\mu_{s})\subset\mathcal{E}^{+}_{\alpha}$ converges to the same (unique)
limit both strongly and vaguely. This in particular implies that if
$A\subset\mathbb{R}^{n}$ is closed (in $\mathbb{R}^{n}$), then the cone
$\mathcal{E}^{+}_{A}:=\mathcal{E}^{+}_{\alpha}\cap\mathfrak{M}^{+}_{A}$ is
strongly complete, $\mathfrak{M}^{+}_{A}$ being vaguely closed.

The inner $\alpha$-Riesz capacity $c_{\alpha}(A)$ of arbitrary
$A\subset\mathbb{R}^{n}$ is defined by the formula222As usual, the infimum
over the empty set is taken to be $+\infty$. We also agree that
$1/(+\infty)=0$ and $1/0=+\infty$.

$c_{\alpha}(A):=1\bigl{/}\inf\,\|\mu\|^{2},$ (2.1)

the infimum being taken over all $\mu\in\mathcal{E}^{+}_{A}$ with
$\mu(\mathbb{R}^{n})=1$. If $c_{\alpha}(A)<\infty$, there exists the unique
solution $\gamma_{A}$ to the minimum $\alpha$-Riesz energy problem over the
class of all $\nu\in\mathcal{E}^{+}_{\alpha}$ such that
$\kappa_{\alpha}\nu\geqslant 1$ n.e. on $A$ (see e.g. [15, Theorem 4.1]). This
$\gamma_{A}$ is said to be the inner $\alpha$-Riesz equilibrium measure for
$A$,333We also refer to the author’s recent work [28] providing a number of
alternative characterizations of the inner capacity $c_{\alpha}(A)$ and the
inner equilibrium measure $\gamma_{A}$, the results in [28] being actually
valid even for quite a large class of general function kernels on locally
compact spaces. and it satisfies the relations

$\displaystyle\gamma_{A}(\mathbb{R}^{n})=\kappa_{\alpha}(\gamma_{A},\gamma_{A})=c_{\alpha}(A)\in[0,\infty),$
(2.2) $\displaystyle\kappa_{\alpha}\gamma_{A}=1\text{ \ n.e.\ on $A$}.$ (2.3)

For closed $A\subset\mathbb{R}^{n}$, the inner $\alpha$-Riesz equilibrium
measure $\gamma_{A}$ can alternatively be characterized as the only measure in
$\mathcal{E}^{+}_{A}$ satisfying (2.3). But if $A$ is not closed, $\gamma_{A}$
may not be concentrated on the set $A$ itself, but on $\overline{A}$, the
closure of $A$ in $\mathbb{R}^{n}$; and moreover, there is in general no
$\nu\in\mathcal{E}^{+}_{A}$ having the property $\kappa_{\alpha}\nu=1$ n.e. on
$A$. Regarding the latter, see [28] (Theorem 1.1(e) and footnote 3).

For reasons of homogeneity,

$c_{\alpha}(A)=0\iff\mathcal{E}^{+}_{A}=\\{0\\}\iff\mathcal{E}^{+}_{K}=\\{0\\}\text{
\ for every compact $K\subset A$}$

(cf. [15, Lemma 2.3.1]). This in turn implies that for any measure
$\mu\in\mathcal{E}^{+}_{\alpha}$ and any $\mu$-measurable set
$A\subset\mathbb{R}^{n}$ with $c_{\alpha}(A)=0$, $A$ is $\mu$-negligible.

Along with the perfectness of the $\alpha$-Riesz kernel, Theorems 2.1 and 2.2
were crucial to the development of both the theory of inner $\alpha$-Riesz
balayage and that of inner $\alpha$-Riesz equilibrium measures (see [24]–[28],
cf. Sects. 3, 4 below).

######  Theorem 2.1.

If a net $(\kappa_{\alpha}\mu_{s})_{s\in S}$, where $(\mu_{s})_{s\in
S}\subset\mathfrak{M}^{+}$, increases pointwise on $\mathbb{R}^{n}$, and is
majorized by $\kappa_{\alpha}\nu$ with some $\nu\in\mathfrak{M}^{+}$, then
there exists $\mu_{0}\in\mathfrak{M}^{+}$ such that
$\kappa_{\alpha}\mu_{s}\uparrow\kappa_{\alpha}\mu_{0}$ pointwise on
$\mathbb{R}^{n}$ and $\mu_{s}\to\mu_{0}$ vaguely (as $s$ increases along $S$).

###### Proof.

If $(\mu_{s})_{s\in S}$ is a sequence, Theorem 2.1 is, in fact, [19, Theorem
3.9] (cf. also [7, 9]). The proof of [19, Theorem 3.9] can be generalized to
the case where $(\mu_{s})_{s\in S}$ is a net, by use of [13, Appendix VIII,
Theorem 2] and [5, Section IV.1, Theorem 1]. ∎

The property of the $\alpha$-Riesz kernels (of order $\alpha\in(0,2]$,
$\alpha<n$) presented in the following theorem (see [19, Theorems 1.27,
1.29]), is known in the literature as the complete maximum principle; for
$q=0$, it is also called the domination principle, and for $\nu=0$, the
Frostman maximum principle.

######  Theorem 2.2.

If $\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu+q$ $\mu$-a.e. ($\mu$-almost
everywhere), where $\mu\in\mathcal{E}^{+}_{\alpha}$, $\nu\in\mathfrak{M}^{+}$,
and $q\in[0,\infty)$, then the same inequality actually holds true on all of
$\mathbb{R}^{n}$.

##  3\. Basic facts of the theory of inner $\alpha$-Riesz balayage

The theory of inner $\alpha$-Riesz balayage of arbitrary
$\mu\in\mathfrak{M}^{+}$ to arbitrary $A\subset\mathbb{R}^{n}$, originated by
the author in [24] (for $\alpha=2$, see the pioneering paper by Cartan [10]),
has recently found a further development in [25]–[28]. The present section as
well as Sect. 4 describes some basic facts from [24]–[28], useful for the
understanding of the results of the current study and the methods applied.

Assume for a moment that a set $A:=F$ is closed, and that a measure
$\mu:=\sigma$ is of finite energy, i.e. $\sigma\in\mathcal{E}^{+}_{\alpha}$.
Based on the facts that the $\alpha$-Riesz kernel is perfect and satisfies the
domination principle, one can prove by generalizing the classical Gauss
variational method (see [8, 10], cf. also [19, Section IV.5.23]) that there
exists $\sigma^{F}\in\mathcal{E}^{+}_{F}$ uniquely determined within
$\mathcal{E}^{+}_{F}$ by the equality
$\kappa_{\alpha}\sigma^{F}=\kappa_{\alpha}\sigma$ n.e. on $F$. This
$\sigma^{F}$ is said to be the $\alpha$-Riesz balayage of
$\sigma\in\mathcal{E}^{+}_{\alpha}$ onto (closed) $F$, and it can
alternatively be characterized as the orthogonal projection of $\sigma$ in the
pre-Hilbert space $\mathcal{E}_{\alpha}$ onto the convex, strongly complete
cone $\mathcal{E}^{+}_{F}$ (see Sect. 2); that is,444Concerning the orthogonal
projection in a pre-Hilbert space, see e.g. [14, Theorem 1.12.3].

$\|\sigma-\sigma^{F}\|=\min_{\nu\in\mathcal{E}^{+}_{F}}\,\|\sigma-\nu\|.$
(3.1)

However, if $A$ is not closed, or if $\mu$ is of infinite energy, there is in
general no measure $\nu$ which would be uniquely determined within
$\mathfrak{M}^{+}_{A}$ by the equality $\kappa_{\alpha}\nu=\kappa_{\alpha}\mu$
n.e. on $A$ (see e.g. Remark 3.5). Nevertheless, a substantial theory of inner
$\alpha$-Riesz balayage of arbitrary $\mu\in\mathfrak{M}^{+}$ to arbitrary
$A\subset\mathbb{R}^{n}$ was developed [24]–[28], and this was performed by
means of several alternative approaches described below.555The outer
$\alpha$-Riesz balayage was investigated by J. Bliedtner and W. Hansen [2] in
the general framework of balayage spaces. See also N.S. Landkof [19, Section
V.1.2], where however it was imposed upon $A$ and $\mu$ certain restrictions,
e.g. that $A\subset\mathbb{R}^{n}$ be Borel while $\mu\in\mathfrak{M}^{+}$ be
bounded.

Given arbitrary $\mu\in\mathfrak{M}^{+}$ and $A\subset\mathbb{R}^{n}$, denote

$\Gamma_{A,\mu}:=\bigl{\\{}\nu\in\mathfrak{M}^{+}:\
\kappa\nu\geqslant\kappa\mu\text{ \ n.e.\ on $A$}\bigr{\\}}.$ (3.2)

The class $\Gamma_{A,\mu}$ is obviously nonempty, for $\mu\in\Gamma_{A,\mu}$,
and it is convex, the latter being clear from the following strengthened
version of countable subadditivity for inner capacity (see [10, p. 253], [15,
p. 158, Remark]).

######  Lemma 3.1.

For arbitrary $A\subset\mathbb{R}^{n}$ and Borel $B_{j}\subset\mathbb{R}^{n}$,
$j\in\mathbb{N}$,

$c_{\alpha}\Bigl{(}\bigcup_{j\in\mathbb{N}}\,A\cap
B_{j}\Bigr{)}\leqslant\sum_{j\in\mathbb{N}}\,c_{\alpha}(A\cap B_{j}).$

######  Definition 3.2.

The inner balayage $\mu^{A}$ of $\mu\in\mathfrak{M}^{+}$ to
$A\subset\mathbb{R}^{n}$ is defined as the measure of minimum potential in the
class $\Gamma_{A,\mu}$, that is, $\mu^{A}\in\Gamma_{A,\mu}$ and

$\kappa_{\alpha}\mu^{A}=\min_{\nu\in\Gamma_{A,\mu}}\,\kappa_{\alpha}\nu.$
(3.3)

This definition is in agreement with Cartan’s classical concept of inner
Newtonian balayage (cf. [10, Section 19, Theorem 1]). Nonetheless, the results
presented below are largely new even for the Newtonian kernel $|x-y|^{2-n}$ on
$\mathbb{R}^{n}$, $n\geqslant 3$.

Denote by $\mathcal{E}^{\prime}_{A}$ the closure of $\mathcal{E}^{+}_{A}$ in
the strong topology on $\mathcal{E}_{\alpha}^{+}$. The class
$\mathcal{E}^{\prime}_{A}$ is convex, for so is $\mathcal{E}^{+}_{A}$; and it
is strongly complete, being a strongly closed subset of the strongly complete
cone $\mathcal{E}^{+}_{\alpha}$ (cf. Sect. 2).

######  Theorem 3.3.

Given arbitrary $\mu\in\mathfrak{M}^{+}$ and $A\subset\mathbb{R}^{n}$, the
inner balayage $\mu^{A}$, introduced by Definition 3.2, exists and is unique.
Furthermore,666Relation (3.4) actually holds true everywhere on $A^{r}$,
$A^{r}\subset\overline{A}$ being the set of all inner $\alpha$-regular points
for $A$ (see (3.10), cf. also (3.12)).

$\displaystyle\kappa_{\alpha}\mu^{A}$ $\displaystyle=\kappa_{\alpha}\mu\text{
\ n.e.\ on\ }A,$ (3.4) $\displaystyle\kappa_{\alpha}\mu^{A}$
$\displaystyle\leqslant\kappa_{\alpha}\mu\text{ \ on\ }\mathbb{R}^{n}.$ (3.5)

The inner balayage $\mu^{A}$ can alternatively be characterized by means of
either of the following (equivalent) assertions:

  * (a)

$\mu^{A}$ is the unique measure in $\mathfrak{M}^{+}$ satisfying the symmetry
relation

$\kappa_{\alpha}(\mu^{A},\sigma)=\kappa_{\alpha}(\sigma^{A},\mu)\text{ \ for
all\/ $\sigma\in\mathcal{E}_{\alpha}^{+}$},$

where $\sigma^{A}$ denotes the only measure in $\mathcal{E}^{\prime}_{A}$ with
$\kappa_{\alpha}\sigma^{A}=\kappa_{\alpha}\sigma$ n.e. on $A$.777For any
$\sigma\in\mathcal{E}^{+}_{\alpha}$ and any $A\subset\mathbb{R}^{n}$, the
measure $\sigma^{A}\in\mathcal{E}^{\prime}_{A}$ having the property
$\kappa_{\alpha}\sigma^{A}=\kappa_{\alpha}\sigma$ n.e. on $A$, exists and is
unique. It is in fact the orthogonal projection of $\sigma$ in the pre-Hilbert
space $\mathcal{E}_{\alpha}$ onto the convex, strongly complete cone
$\mathcal{E}^{\prime}_{A}$; that is (compare with (3.1)),
$\|\sigma-\sigma^{A}\|=\min_{\nu\in\mathcal{E}^{\prime}_{A}}\,\|\sigma-\nu\|.$
Alternatively, $\sigma^{A}$ is uniquely characterized within
$\mathfrak{M}^{+}$ by extremal property (3.3) with $\mu:=\sigma$.

  * (b)

$\mu^{A}$ is the unique measure in $\mathfrak{M}^{+}$ satisfying either of the
two limit relations

$\displaystyle\mu_{j}^{A}\to\mu^{A}\text{ \ vaguely in\/ $\mathfrak{M}^{+}$
as\/ $j\to\infty$},$
$\displaystyle\kappa_{\alpha}\mu_{j}^{A}\uparrow\kappa_{\alpha}\mu^{A}\text{ \
pointwise on\/ $\mathbb{R}^{n}$ as\/ $j\to\infty$},$

where $(\mu_{j})$ is an arbitrary sequence of
$\mu_{j}\in\mathcal{E}^{+}_{\alpha}$ having the property888Such
$\mu_{j}\in\mathcal{E}^{+}_{\alpha}$, $j\in\mathbb{N}$, do exist; they can be
defined, for instance, by means of the formula
$\kappa_{\alpha}\mu_{j}:=\min\,\bigl{\\{}\kappa_{\alpha}\mu,\,j\kappa_{\alpha}\lambda\bigr{\\}},$
$\lambda\in\mathcal{E}^{+}_{\alpha}$ being fixed (see e.g. [19, p. 272] or
[10, p. 257, footnote]). Here we have used the fact that for any
$\mu_{1},\mu_{2}\in\mathfrak{M}^{+}$, there is $\mu_{0}\in\mathfrak{M}^{+}$
such that
$\kappa_{\alpha}\mu_{0}:=\min\,\\{\kappa_{\alpha}\mu_{1},\,\kappa_{\alpha}\mu_{2}\\}$
[19, Theorem 1.31].

$\kappa_{\alpha}\mu_{j}\uparrow\kappa_{\alpha}\mu\text{ \ pointwise on\/
$\mathbb{R}^{n}$ as\/ $j\to\infty$},$ (3.6)

whereas $\mu_{j}^{A}$ denotes the only measure in $\mathcal{E}^{\prime}_{A}$
with $\kappa_{\alpha}\mu_{j}^{A}=\kappa_{\alpha}\mu_{j}$ n.e. on $A$.
(Regarding the existence and uniqueness of this $\mu_{j}^{A}$, see footnote
7).

######  Remark 3.4.

The inner balayage $\mu^{A}$ is in general not concentrated on the set $A$
itself, but on its closure $\overline{A}$, and this occurs even for the
Newtonian kernel on $\mathbb{R}^{n}$, $n\geqslant 3$, and
$A:=B_{r}:=\\{|x|<r\\}$, $r\in(0,\infty)$. Indeed, for any
$\mu\in\mathfrak{M}^{+}_{B_{r}^{c}}$, we have $S(\mu^{B_{r}})=\\{|x|=r\\}$
(see [25, Theorems 4.1, 5.1]), and hence actually $S(\mu^{B_{r}})\cap
B_{r}=\varnothing$.

######  Remark 3.5.

Assume for a moment that $\mu\in\mathcal{E}^{+}_{\alpha}$. As noted in Theorem
3.3 and footnote 7, the inner balayage $\mu^{A}$ is then the only measure in
$\mathcal{E}^{\prime}_{A}$ satisfying (3.4). This in turn implies that there
is in general no $\nu\in\mathcal{E}^{+}_{A}$ with
$\kappa_{\alpha}\nu=\kappa_{\alpha}\mu$ n.e. on $A$. Indeed, if there were
such $\nu$, then it would necessarily serve as $\mu^{A}$; which however is in
general impossible, for $\mu^{A}$ may not be concentrated on $A$ (see Remark
3.4).

######  Corollary 3.6.

For any $\mu\in\mathfrak{M}^{+}$ and any $A\subset\mathbb{R}^{n}$, the inner
balayage $\mu^{A}$ is of minimum total mass in the class $\Gamma_{A,\mu}$,
that is,

$\mu^{A}(\mathbb{R}^{n})=\min_{\nu\in\Gamma_{A,\mu}}\,\nu(\mathbb{R}^{n}).$
(3.7)

###### Proof.

Since $\mu^{A}\in\Gamma_{A,\mu}$, we only need to show that
$\mu^{A}(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$ for all
$\nu\in\Gamma_{A,\mu}$, which however follows directly from definition (3.2)
by use of the (classical) principle of positivity of mass (see Theorem 1.1). ∎

######  Remark 3.7.

However, extremal property (3.7) cannot serve as an alternative
characterization of inner balayage, for it does not determine $\mu^{A}$
uniquely within $\Gamma_{A,\mu}$. Indeed, consider a closed proper subset $A$
of $\mathbb{R}^{n}$ that is not $\alpha$-thin at infinity (take, for instance,
$A:=\\{|x|\geqslant 1\\}$). Then for any $\mu\in\mathfrak{M}^{+}_{A^{c}}$,

$\mu^{A}\neq\mu\text{ \ and \ }\mu^{A}(\mathbb{R}^{n})=\mu(\mathbb{R}^{n}),$
(3.8)

the former relation being obvious, and the latter following from Theorem
4.2(iii). Noting that $\mu,\mu^{A}\in\Gamma_{A,\mu}$ while $\Gamma_{A,\mu}$ is
convex, we conclude by combining (3.7) with (3.8) that there are actually in
$\Gamma_{A,\mu}$ infinitely many measures of minimum total mass, for so is
every measure of the form $a\mu+b\mu^{A}$, where $a,b\in[0,1]$ and $a+b=1$.

######  Corollary 3.8.

Given $\mu\in\mathfrak{M}^{+}$ and $A\subset\mathbb{R}^{n}$,

$\kappa_{\alpha}(\mu^{A},\nu)=\kappa_{\alpha}(\mu,\nu^{A})\text{ \ for all\/
$\nu\in\mathfrak{M}^{+}$}.$ (3.9)

###### Proof.

Fix $\mu,\nu\in\mathfrak{M}^{+}$, and choose a sequence
$(\mu_{j})\subset\mathcal{E}^{+}_{\alpha}$ satisfying (3.6). By Theorem 3.3
(see (a) and (b)), $(\kappa_{\alpha}\mu_{j}^{A})$ increases pointwise on
$\mathbb{R}^{n}$ to $\kappa_{\alpha}\mu^{A}$, whereas

$\int\kappa_{\alpha}\mu_{j}^{A}\,d\nu=\int\kappa_{\alpha}\mu_{j}\,d\nu^{A}\text{
\ for all\ $j$}.$

Applying the monotone convergence theorem [5, Section IV.1, Theorem 3] to each
of these two integrals, we obtain (3.9). ∎

######  Remark 3.9.

It follows from Theorem 3.3(a) and Corollary 3.8 that, if for a given
$\mu\in\mathfrak{M}^{+}$, there exists $\zeta\in\mathfrak{M}^{+}$ having the
property

$\kappa_{\alpha}(\zeta,\nu)=\kappa_{\alpha}(\mu,\nu^{A})\text{ \ for all
$\nu\in\mathfrak{M}^{+}$,}$

then necessarily $\zeta=\mu^{A}$. Actually, this characteristic property of
the inner balayage $\mu^{A}$ needs only to be verified for certain countably
many $\nu_{j}\in\mathcal{E}^{+}_{\alpha}$ that are independent of the choice
of $\mu\in\mathfrak{M}^{+}$.999This result has recently been extended to inner
balayage on a locally compact space, see [26]. This fact is implied by the
observation that there exist countably many
$\nu_{j}\in\mathcal{E}^{+}_{\alpha}$ whose potentials $\kappa_{\alpha}\nu_{j}$
form a dense subset of $C_{0}(\mathbb{R}^{n})$, see [25, Lemmas 3.1, 3.2].

Given $A\subset\mathbb{R}^{n}$, denote by $\mathfrak{C}_{A}$ the upward
directed set of all compact subsets $K$ of $A$, where $K_{1}\leqslant K_{2}$
if and only if $K_{1}\subset K_{2}$. If a net
$(x_{K})_{K\in\mathfrak{C}_{A}}\subset Y$ converges to $x_{0}\in Y$, $Y$ being
a topological space, then we shall indicate this fact by writing

$x_{K}\to x_{0}\text{ \ in $Y$ as $K\uparrow A$}.$

The following theorem (cf. [24, Theorem 4.5]), analyzing the convergence of
inner swept measures and their potentials under the exhaustion of
$A\subset\mathbb{R}^{n}$ by compact subsets $K\subset A$, justifies the term
‘inner’ balayage.

######  Theorem 3.10.

For any $\mu\in\mathfrak{M}^{+}$ and any $A\subset\mathbb{R}^{n}$,

$\displaystyle\mu^{K}\to\mu^{A}\text{ \ vaguely in\/ $\mathfrak{M}^{+}$ as\/
$K\uparrow A$},$
$\displaystyle\kappa_{\alpha}\mu^{K}\uparrow\kappa_{\alpha}\mu^{A}\text{ \
pointwise on\/ $\mathbb{R}^{n}$ as\/ $K\uparrow A$}.$

If now $\mu\in\mathcal{E}^{+}_{\alpha}$, then also

$\mu^{K}\to\mu^{A}\text{ \ strongly in\/ $\mathcal{E}^{+}_{\alpha}$ as\/
$K\uparrow A$}.$

A point $y\in\mathbb{R}^{n}$ is said to be inner $\alpha$-regular for $A$ if
$\varepsilon_{y}=(\varepsilon_{y})^{A}=:\varepsilon_{y}^{A}$,
$\varepsilon_{y}$ being the unit Dirac measure at
$y$;101010$\varepsilon_{y}^{A}$ is said to be the (fractional) inner
$\alpha$-harmonic measure of $A\subset\mathbb{R}^{n}$ at $y\in\mathbb{R}^{n}$.
Being a natural generalization of the classical concept of ($2$-)harmonic
measure [2, 3, 4, 19], $\varepsilon_{y}^{A}$ serves as the main tool in
solving the generalized Dirichlet problem for $\alpha$-harmonic functions.
Besides, due to the integral representation formula
$\mu^{A}=\int\varepsilon_{y}^{A}\,d\mu(y)$ [25, Theorem 5.1], the inner
$\alpha$-harmonic measure $\varepsilon_{y}^{A}$ is a powerful tool in the
investigation of the inner balayage $\mu^{A}$ for arbitrary $\mu$ (see [25]).
the set of all those $y$ is denoted by $A^{r}$. Then
$A^{r}\subset\overline{A}$, since obviously
$\varepsilon_{x}^{A}\in\mathcal{E}^{+}_{\alpha}$ for all
$x\notin\overline{A}$. The other points of $\overline{A}$, i.e.

$y\in\overline{A}\setminus A^{r}=:A_{I},$

are said to be inner $\alpha$-irregular for $A$. As seen from (3.9) with
$\nu:=\varepsilon_{y}$,

$y\in A^{r}\iff\kappa_{\alpha}\mu^{A}(y)=\kappa_{\alpha}\mu(y)\text{ \ for
all\ }\mu\in\mathfrak{M}^{+}.$ (3.10)

It is also worth noting that for any $A\subset\mathbb{R}^{n}$, the sets
$A^{r}$ and $A_{I}$ are Borel measurable [25, Theorem 5.2], and hence
capacitable.

By the Wiener type criterion [24, Theorem 6.4],

$y\notin
A^{r}\iff\sum_{j\in\mathbb{N}}\,\frac{c_{\alpha}(A_{j})}{q^{j(n-\alpha)}}<\infty,$
(3.11)

where $q\in(0,1)$ and $A_{j}:=A\cap\\{x\in\mathbb{R}^{n}:\
q^{j+1}<|x-y|\leqslant q^{j}\\}$, while by the Kellogg–Evans type theorem [24,
Theorem 6.6],111111Observe that both (3.11) and (3.12) refer to inner
capacity; compare with the Kellogg–Evans and Wiener type theorems established
for outer balayage (see e.g. [2, 6, 10, 13]). Regarding (3.12), we also note
that the whole set $A_{I}$ may be of nonzero capacity [19, Section V.4.12].

$c_{\alpha}(A\cap A_{I})=0.$ (3.12)

In particular, (3.11) implies that $A_{I}\subset\partial A$, where $\partial
A$ denotes the boundary of $A$ in the Euclidean topology on $\mathbb{R}^{n}$.

##  4\. Basic facts of the theory of inner $\alpha$-Riesz equilibrium measures

This section reviews some basic facts of the theory of inner $\alpha$-Riesz
equilibrium measures, developed in [24, 25, 28]. The inner equilibrium measure
$\gamma_{A}$ of $A\subset\mathbb{R}^{n}$ is understood in an extended sense
where its energy $\kappa_{\alpha}(\gamma_{A},\gamma_{A})$ as well as its total
mass $\gamma_{A}(\mathbb{R}^{n})$ may be infinite (compare with (2.2)).

For arbitrary $A\subset\mathbb{R}^{n}$, define

$\Gamma_{A}:=\bigl{\\{}\nu\in\mathfrak{M}^{+}:\ \kappa_{\alpha}\nu\geqslant
1\text{ \ n.e.\ on $A$}\bigr{\\}}.$ (4.1)

######  Definition 4.1.

A measure $\gamma_{A}$ is said to be the inner $\alpha$-Riesz equilibrium
measure of $A\subset\mathbb{R}^{n}$ if it is of minimum potential in
$\Gamma_{A}$, that is, if $\gamma_{A}\in\Gamma_{A}$ and121212In view of
(permanent) assumption (1.1), the inner $\alpha$-Riesz equilibrium measure
$\gamma_{A}$ does not exist if there is no $\nu\in\mathfrak{M}^{+}$ with
$\kappa_{\alpha}\nu\geqslant 1$ n.e. on $A$. This implication can actually be
reversed, and hence $\gamma_{A}$ exists if and only if
$\Gamma_{A}\neq\varnothing$, see Theorem 4.2(ii).

$\kappa_{\alpha}\gamma_{A}=\min_{\nu\in\Gamma_{A}}\,\kappa_{\alpha}\nu.$

It follows easily by use of [19, Theorem 1.12] that the inner $\alpha$-Riesz
equilibrium measure $\gamma_{A}$ is unique (if exists). Theorem 4.2 below
provides a number of equivalent conditions, each of which is necessary and
sufficient for the existence of this $\gamma_{A}$. The approach applied is
based on a close interaction between the concept of inner equilibrium measure
and that of inner balayage, described by means of equality (4.5) with the
Kelvin transformation involved.

For every $y\in\mathbb{R}^{n}$, define the inversion $J_{y}$ with respect to
the sphere $S(y,1):=\\{x\in\mathbb{R}^{n}:\ |x-y|=1\\}$ mapping each point
$x\neq y$ to the point $x^{*}=J_{y}(x)$ on the ray through $x$ issuing from
$y$ which is uniquely determined by

$|x-y|\cdot|x^{*}-y|=1.$

This is a homeomorphism of $\mathbb{R}^{n}\setminus\\{y\\}$ onto itself having
the property

$|x^{*}-z^{*}|=\frac{|x-z|}{|x-y||z-y|}\text{ \ for all\
}x,z\in\mathbb{R}^{n}\setminus\\{y\\}.$ (4.2)

If now $A\subset\mathbb{R}^{n}$ is given, then for any $q\in(0,1)$ and any
$A_{j}$ appearing in the Wiener type criterion (3.11) of inner
$\alpha$-regularity,

$q^{-2j(n-\alpha)}c_{\alpha}(A_{j})\leqslant c_{\alpha}(A_{j}^{*})\leqslant
q^{-(2j+2)(n-\alpha)}c_{\alpha}(A_{j}),\ j\in\mathbb{N},$

where $A_{j}^{*}:=J_{y}(A_{j})$. This follows from (4.2) by use of [19, Remark
to Theorem 2.9].

For every $\nu\in\mathfrak{M}^{+}$ with $\nu(\\{y\\})=0$, define the Kelvin
transform $\nu^{*}=\mathcal{K}_{y}\nu\in\mathfrak{M}^{+}$ by means of the
formula (see [19, Section IV.5.19])

$d\nu^{*}(x^{*})=|x-y|^{\alpha-n}\,d\nu(x),\text{ \ where\
}x^{*}=J_{y}(x)\in\mathbb{R}^{n}.$ (4.3)

Noting that $(\nu^{*})^{*}=\nu$, we derive from (4.2) and (4.3) that

$\displaystyle\nu(\mathbb{R}^{n})=\kappa_{\alpha}\nu^{*}(y),$
$\displaystyle\kappa_{\alpha}(\nu^{*},\nu^{*})=\kappa_{\alpha}(\nu,\nu),$
(4.4)
$\displaystyle\kappa_{\alpha}\nu^{*}(x^{*})=|x-y|^{n-\alpha}\kappa_{\alpha}\nu(x)\text{
\ for all\ }x^{*}\in\mathbb{R}^{n}.$

The proof of the following theorem is based on the theory of inner balayage,
reviewed in Sect. 3, as well as on the above-quoted elementary properties of
the inversion and the Kelvin transformation (see [25, Theorems 2.1, 2.2, 5.1]
for details).

######  Theorem 4.2.

For arbitrary $A\subset\mathbb{R}^{n}$, the following (i)–(v) are equivalent.

  * (i)

There exists the inner $\alpha$-Riesz equilibrium measure $\gamma_{A}$ for
$A$, uniquely determined by Definition 4.1.

  * (ii)

There exists $\nu\in\mathfrak{M}^{+}$ with

$\operatorname*{ess\,inf}_{x\in A}\,\kappa_{\alpha}\nu(x)>0,$

where $x$ ranges over all of $A$ except for a subset of inner capacity zero.

  * (iii)

There exists $\mu\in\mathfrak{M}^{+}$ having the property131313In general,
$\mu^{Q}(\mathbb{R}^{n})\leqslant\mu(\mathbb{R}^{n})$, where
$\mu\in\mathfrak{M}^{+}$ and $Q\subset\mathbb{R}^{n}$ are arbitrary. This
follows e.g. from (3.5) by use of the (classical) principle of positivity of
mass (see Theorem 1.1).

$\mu^{A}(\mathbb{R}^{n})<\mu(\mathbb{R}^{n}).$

  * (iv)

For some (equivalently, every) $y\in\mathbb{R}^{n}$,

$\sum_{j\in\mathbb{N}}\,\frac{c_{\alpha}(A_{j})}{q^{j(n-\alpha)}}<\infty,$

where $q\in(1,\infty)$ and $A_{j}:=A\cap\\{x\in\mathbb{R}^{n}:\
q^{j}\leqslant|x-y|<q^{j+1}\\}$.

  * (v)

For some (equivalently, every) $y\in\mathbb{R}^{n}$, the inner
$\alpha$-harmonic measure $\varepsilon_{y}^{A_{y}^{*}}$ is
$c_{\alpha}$-absolutely continuous.

If these (i)–(v) hold true, then for every $y\in\mathbb{R}^{n}$, the Kelvin
transform $(\gamma_{A})^{*}=\mathcal{K}_{y}\gamma_{A}$ of the inner
equilibrium measure $\gamma_{A}$ for the set $A$ is actually the inner
$\alpha$-harmonic measure $\varepsilon_{y}^{A_{y}^{*}}$ for the inverse
$A_{y}^{*}=J_{y}(A)$. That is,

$\varepsilon_{y}^{A_{y}^{*}}=(\gamma_{A})^{*}.$ (4.5)

######  Definition 4.3 (see [25, Definition 2.1]).

A set $A\subset\mathbb{R}^{n}$ is said to be inner $\alpha$-thin at infinity
if any of (equivalent) assertions (i)–(v) in Theorem 4.2 holds true. Thus $A$
is inner $\alpha$-thin at infinity if and only if some (equivalently, every)
point $y\in\mathbb{R}^{n}$ is inner $\alpha$-irregular for $A_{y}^{*}$, the
inverse of $A$ with respect to the sphere $S(y,1)$.141414The concept of inner
$\alpha$-thinness of a set at infinity, and particularly its alternative
characterization provided by Theorem 4.2(iii), has already found a number of
applications to minimum $\alpha$-Riesz energy problems for condensers, see
e.g. [17], [21]–[23].

######  Remark 4.4.

The concept of inner $\alpha$-thinness at infinity thus introduced actually
coincides with that of $\alpha$-thinness at infinity, defined by T. Kurokawa
and Y. Mizuta [18, Definition 3.1]. To validate this claim, it is enough to
show that the concept of capacity used in [18] (see p. 534 for its definition)
coincides with that of inner capacity, defined by (2.1); which however can be
concluded from [28, Theorem 4.3]. Also observe that in the case where
$\alpha=2$ and $A$ is Borel, the above concept of inner $2$-thinness at
infinity coincides with that of outer $2$-thinness at infinity, introduced by
J.L. Doob [13, pp. 175–176].

For the following theorem we refer to [25, Theorem 2.3] (compare with [19,
Section V.2.8], where $A$ was Borel).

######  Theorem 4.5.

For arbitrary $A\subset\mathbb{R}^{n}$, the following (i1)–(iii1) are
equivalent.

  * (i1)

The inner $\alpha$-Riesz capacity of $A$ is finite:

$c_{\alpha}(A)<\infty.$

  * (ii1)

For some (equivalently, every) $y\in\mathbb{R}^{n}$,

$\sum_{j\in\mathbb{N}}\,c_{\alpha}(A_{j})<\infty,$

where $q\in(1,\infty)$ and $A_{j}:=A\cap\\{x\in\mathbb{R}^{n}:\
q^{j}\leqslant|x-y|<q^{j+1}\\}$.

  * (iii1)

For some (equivalently, every) $y\in\mathbb{R}^{n}$,

$\varepsilon_{y}^{A_{y}^{*}}\in\mathcal{E}^{+}_{\alpha}.$

######  Corollary 4.6.

The following two conclusions (a) and (b) are obtained by comparing Theorems
4.2 and 4.5.

  * (a)

If $A\subset\mathbb{R}^{n}$ is not $\alpha$-thin at infinity, then necessarily
$c_{\alpha}(A)=\infty$.

  * (b)

There exists $A_{0}\subset\mathbb{R}^{n}$ which is $\alpha$-thin at infinity,
but nonetheless, $c_{\alpha}(A_{0})=\infty$.

######  Example 4.7.

For instance, the set $F_{2}\subset\mathbb{R}^{3}$, defined by means of
formulae (1.3) and (1.4), is $2$-thin at infinity; whereas its Newtonian
capacity is finite if and only if $s>1$, where $s$ is the parameter involved
in (1.4).

Using (4.4) and (4.5), we conclude from Theorem 4.5 that the inner
$\alpha$-Riesz equilibrium measure $\gamma_{A}$ has finite energy if and only
if $c_{\alpha}(A)<\infty$; and in the affirmative case it can alternatively be
characterized, for instance, as the only measure in $\mathcal{E}^{\prime}_{A}$
having the property $\kappa_{\alpha}\gamma_{A}=1$ n.e. $A$. Regarding the
latter, see [28, Theorem 9.1].

In the case where the inner equilibrium measure $\gamma_{A}$ still exists,
although its energy may now be infinite, $\gamma_{A}$ can be described as
follows (see [24, Sections 5, 6]).

######  Theorem 4.8.

For any $A\subset\mathbb{R}^{n}$ that is inner $\alpha$-thin at infinity, the
inner $\alpha$-Riesz equilibrium measure $\gamma_{A}$ has the
properties151515Thus $\kappa_{\alpha}\gamma_{A}=1$ n.e. on $A$, see (3.12) and
(4.6).

$\displaystyle\kappa_{\alpha}\gamma_{A}=1\text{ \ on\/ $A^{r}$},$ (4.6)
$\displaystyle\kappa_{\alpha}\gamma_{A}\leqslant 1\text{ \ on\/
$\mathbb{R}^{n}$}.$

Furthermore, $\gamma_{A}$ can be characterized as the unique measure in
$\mathfrak{M}^{+}$ satisfying either of the two limit relations

$\displaystyle\gamma_{K}\to\gamma_{A}\text{ \ vaguely in\/ $\mathfrak{M}^{+}$
as\/ $K\uparrow A$},$ (4.7)
$\displaystyle\kappa_{\alpha}\gamma_{K}\uparrow\kappa_{\alpha}\gamma_{A}\text{
\ pointwise on\/ $\mathbb{R}^{n}$ as\/ $K\uparrow A$},$ (4.8)

where $\gamma_{K}$ denotes the only measure in $\mathcal{E}^{+}_{K}$ with
$\kappa_{\alpha}\gamma_{K}=1$ n.e. on $K$.

######  Remark 4.9.

Alternatively, (4.7) and (4.8) will follow from Theorem 3.10 once we show that
for any given $Q\subset A$,

$\gamma_{Q}=(\gamma_{A})^{Q}.$

To this end, it is enough to prove that $\Gamma_{Q}=\Gamma_{Q,\gamma_{A}}$
(cf. Definitions 3.2, 4.1). This however is obvious in view of the fact that
for any given $\nu\in\Gamma_{Q,\gamma_{A}}\cup\Gamma_{Q}$,

$\kappa_{\alpha}\nu\geqslant 1=\kappa_{\alpha}\gamma_{A}\text{ \ n.e.\ on
$Q$},$

which in turn is derived from (3.2) (with $A:=Q$ and $\mu:=\gamma_{A}$),
(4.1), and (4.6) by making use of Lemma 3.1.

##  5\. Proofs of Theorems 1.2, 1.4, and 1.5

###  5.1. Proof of Theorem 1.2

Given $\mu,\nu\in\mathfrak{M}^{+}$, assume that

$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu\text{ \ n.e.\ on $A$},$ (5.1)

where $A\subset\mathbb{R}^{n}$ is not inner $\alpha$-thin at infinity
(Definition 4.3). By Theorem 4.2(iii),

$\mu^{A}(\mathbb{R}^{n})=\mu(\mathbb{R}^{n}),$ (5.2)

$\mu^{A}$ being the inner balayage of $\mu$ to $A$. Noting from (5.1) that
$\nu\in\Gamma_{A,\mu}$, where $\Gamma_{A,\mu}$ was introduced by (3.2), we
conclude from Definition 3.2 that

$\kappa_{\alpha}\mu^{A}\leqslant\kappa_{\alpha}\nu\text{ \ everywhere on
$A$},$

and hence, by applying Theorem 1.1, that

$\mu^{A}(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n}).$

Combining this with (5.2) proves
$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n})$, which was the claim.

###  5.2. Proof of Theorem 1.4

Let $A\subset\mathbb{R}^{n}$ be inner $\alpha$-thin at infinity. Then, by
virtue of Definition 4.3 and Theorem 4.2(iii), one can choose
$\mu_{0}\in\mathfrak{M}^{+}$ so that

$\mu_{0}^{A}(\mathbb{R}^{n})<\mu_{0}(\mathbb{R}^{n}).$ (5.3)

But according to (3.4) applied to $\mu_{0}$,

$\kappa_{\alpha}\mu_{0}^{A}=\kappa_{\alpha}\mu_{0}\text{ \ n.e.\ on $A$},$

which together with (5.3) validates the theorem with $\mu_{0}$, chosen above,
and $\nu_{0}:=\mu_{0}^{A}$.

###  5.3. Proof of Theorem 1.5

Given $A\subset\mathbb{R}^{n}$ and $c_{\alpha}$-absolutely continuous measures
$\mu,\nu\in\mathfrak{M}^{+}_{A}$ having property (5.1), we aim at showing that
then necessarily

$\mu(\mathbb{R}^{n})\leqslant\nu(\mathbb{R}^{n}).$

The set $A^{c}$ being $\xi$-negligible for any $\xi\in\mathfrak{M}^{+}_{A}$,
this is equivalent to the inequality

$\int 1_{A}\,d\mu\leqslant\int 1_{A}\,d\nu,$ (5.4)

where $1_{A}$ denotes the indicator function of $A$.

We can certainly assume that the set $A$ is inner $\alpha$-thin at infinity,
for if not, the claim holds by virtue of Theorem 1.2. According to Theorem
4.2, then there exists the inner $\alpha$-Riesz equilibrium measure
$\gamma_{A}$, uniquely determined by Definition 4.1.

For any compact $K\subset A$, we obtain by Fubini’s theorem

$\int\kappa_{\alpha}\gamma_{K}\,d\mu=\int\kappa_{\alpha}\mu\,d\gamma_{K}\leqslant\int\kappa_{\alpha}\nu\,d\gamma_{K}=\int\kappa_{\alpha}\gamma_{K}\,d\nu,$
(5.5)

where $\gamma_{K}\in\mathcal{E}^{+}_{K}$ denotes the equilibrium measure on
$K$. (The inequality in (5.5) is valid in view of the fact that
$\kappa_{\alpha}\mu\leqslant\kappa_{\alpha}\nu$ holds n.e. on $K$, hence
$\gamma_{K}$-a.e., every Borel $E\subset\mathbb{R}^{n}$ with $c_{\alpha}(E)=0$
being $\xi$-negligible for any $\xi\in\mathcal{E}^{+}_{\alpha}$, see Sect. 2.)

According to (4.8), the net
$(\kappa_{\alpha}\gamma_{K})_{K\in\mathfrak{C}_{A}}$ of positive, lower
semicontinuous functions increases pointwise on $\mathbb{R}^{n}$ to
$\kappa_{\alpha}\gamma_{A}$. Applying [5, Section IV.1, Theorem 1] to the
first and the last integrals in (5.5), we therefore get

$\int\kappa_{\alpha}\gamma_{A}\,d\mu\leqslant\int\kappa_{\alpha}\gamma_{A}\,d\nu.$

This implies (5.4), because $\kappa_{\alpha}\gamma_{A}=1$ holds true n.e. on
$A$ (see (4.6) and (3.12)), hence $(\mu+\nu)$-a.e. To verify the latter,
observe that $N:=A\cap\\{\kappa_{\alpha}\gamma_{A}<1\\}$ is
$(\mu+\nu)$-measurable, for so is the set $A$, $\mu+\nu$ being concentrated on
$A$. Since $c_{\alpha}(N)=0$ while $\mu+\nu$ is $c_{\alpha}$-absolutely
continuous, the set $N$ must be $(\mu+\nu)$-negligible.

This completes the proof of Theorem 1.5. With regard to Remark 1.6, note that
in the case $A\cap A_{I}=\varnothing$, the $c_{\alpha}$-absolute continuity of
$\mu$ and $\nu$ is unnecessary for the validity of the above proof, for then,
again by (4.6), $\kappa_{\alpha}\gamma_{A}=1$ everywhere on $A$.

##  6\. Acknowledgements

The author is deeply indebted to Bent Fuglede for reading and commenting on
the manuscript.

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# Relational Artificial Intelligence

Virginia Dignum Corresponding Author: Virginia Dignum, Umeå University,
Sweden<EMAIL_ADDRESS>
This paper is a modified and extended version of my paper ‘Responsible
Artificial Intelligence: recommendations and lessons learned’, to appear in
the book ‘Responsible AI in Africa’, edited by Damian Okaibedi Eke, Kutoma
Wakunuma and Simisola Akintoye. Computing Science Department, Umeå University,
Sweden

###### Abstract

The impact of Artificial Intelligence does not depend only on fundamental
research and technological developments, but for a large part on how these
systems are introduced into society and used in everyday situations. Even
though AI is traditionally associated with rational decision making,
understanding and shaping the societal impact of AI in all its facets requires
a relational perspective. A rational approach to AI, where computational
algorithms drive decision making independent of human intervention, insights
and emotions, has shown to result in bias and exclusion, laying bare societal
vulnerabilities and insecurities.

A relational approach, that focus on the relational nature of things, is
needed to deal with the ethical, legal, societal, cultural, and environmental
implications of AI. A relational approach to AI recognises that objective and
rational reasoning cannot does not always result in the ‘right’ way to proceed
because what is ‘right’ depends on the dynamics of the situation in which the
decision is taken, and that rather than solving ethical problems the focus of
design and use of AI must be on asking the ethical question.

In this position paper, I start with a general discussion of current
conceptualisations of AI followed by an overview of existing approaches to
governance and responsible development and use of AI. Then, I reflect over
what should be the bases of a social paradigm for AI and how this should be
embedded in relational, feminist and non-Western philosophies, in particular
the Ubuntu philosophy.

##  1 Introduction

As Artificial Intelligence (AI) is increasingly impacting many aspects of
life, the awareness that it has the potential to impact our lives and our
world as no other technology has done before is rightfully raising many
questions concerning its ethical, legal, societal and economical effects.
Moreover, whereas the dangers and risks of application of AI without due
consideration of its societal, ethical or legal impact, are increasingly
acknowledged, the potential of AI to contribute to human and societal well-
being cannot be dismissed. But AI means different things to different people.

Ensuring the responsible development and use of AI is becoming a main
direction in AI research and practice. Governments, corporations and
international organisations alike are coming forward with proposals and
declarations of their commitment to an accountable, responsible, transparent
approach to AI, where human values and ethical principles are leading. A
comprehensive study of the role of AI in achieving the Sustainable Development
Goals111https://sdgs.un.org/ in which I participated, concluded that it has
the potential to shape the delivery of all 17 goals, contributing positively
to 134 targets across all the goals, but it may also inhibit 59 targets [30].
Currently, there are over 600 AI-related policy recommendations, guidelines or
strategy reports, which have been released by prominent intergovernmental
organisations, professional bodies, national-level committees and other public
organisations, non-governmental, and private for-profit companies. A recent
study of the global landscape of AI ethics guidelines shows that there is a
global convergence around five ethical principles: Transparency, Justice and
Fairness, Non-Maleficence, Responsibility, and Privacy [18].

These are much-needed efforts, but still much work is needed to ensure that
all AI is developed and used in responsible ways that contribute to trust and
well-being. Nevertheless, even though organisations agree on the need to
consider ethical, legal and societal principles, how these are interpreted and
applied in practice, varies significantly across the different recommendation
documents.

At the same time, the growing hype around ‘AI’ is blurring its definition and
shoving into the same heap concepts and applications of many different sorts.
A hard needed first step in the responsible development and use of AI is to
ensure a proper AI narrative, one that demystifies its capabilities, minimises
both overselling and underselling of AI-driven solutions, and that enables
wide and inclusive participation in the discussion on the role of AI in
society. Understanding the capabilities and addressing the risks of AI,
requires that all of us, from developers to policy-makers, from provides to
end-users and bystanders, have a clear understanding of what AI is, how it is
applied and what are the opportunities and risks involved.

##  2 What is AI and why should we care?

Without a proper understanding of what AI is and what it can, and cannot, do,
all the efforts towards governance, regulation and responsible use of AI have
the risk to become void. Current AI narratives bring forward benefits and
risks and describe AI in many different ways, from the obvious next step in
digitisation to some kind of magic. If the ‘business as usual’ narrative is
detrimental of innovation and contributes to the maintenance of status quo,
the ‘magic’ narrative, well fed from science fiction, Hollywood and the
popular press, often supports a feeling that nothing can be done against such
an all-knowing entity that rules over us in possibly unexpected ways, either
solving all our problems, or destroying the world in the process. In both
cases, the danger is that the message is that little can be done against the
risks and challenges of AI.

The reality is, as usual, somewhere in the middle. In the following, I briefly
describe some of the ways AI is often misunderstood, and conclude with a
reflection of the significance to the current efforts towards AI governance.

###  2.1 AI is not intelligent

John McCarthy, who originally coined the term Artificial Intelligence, defined
it as “the study and design of intelligent agents”. In this definition, which
is still one of the most common definitions of AI, the concept of intelligence
refers to the ability of computers to perform tasks commonly associated with
intelligent beings, i.e. humans or other non-human animals.

Currently, AI is mostly associated with Machine Learning (ML) but the field of
AI includes many other techniques222Chapter 2 in my book ‘Responsible
Artificial Intelligence’ gives a short, non-technical overview of different
techniques [9].. Machine Learning, and in particular, Neural Networks, or Deep
Learning, is a subset of AI techniques that uses statistical methods to enable
computers to perceive some characteristics of their environment. Current
techniques are particularly efficient in perceiving images, written or spoken
text, as well as the many applications of structured data. By analysing many
thousands of examples (typically a few million), the system is able to
identify commonalities in these examples, which then enable it to interpret
data that it has never seen before, which is often referred to as prediction.

The question remains of what is human (or animal) intelligence, and whether
intelligence can be achieved by correlation techniques [19]. However, there is
no single accepted definition of intelligence, even though in cognitive
science, intelligence is associated with the ability of the mind to reach
correct conclusions about what is true and what is false, and about how to
solve problems [6]. As best, we can say that intelligence is a multifaceted
concept. Psychologists debate on issues such as types of intelligence, the
role of nature versus nurture in intelligence, how intelligence is represented
in the brain, and the meaning of group differences in intelligence. Major
theories include Sternberg’s triarchic theory, Gardner’s theory of multiple
intelligences [17], and Piaget’s theory of development [25]. Many characterise
human intelligence as more than an analytical process and to include creative,
practical and other abilities. These abilities, for a large part associated
with socio-cultural background and context, are far from being possible to be
replicated by AI systems, even if these may approach analytical intelligence
for some (simple) tasks.

###  2.2 AI is not magic

AI is not magic and it is not only artificial. By itself, it will not solve
all our problems, nor can it exist without the use of natural resources and
the work and efforts of legions of people. In a recent book, ‘The Atlas of AI’
[7], Kate Crawford describes the field as a collection of maps that enable the
reader to traverse places, their relations and their impact on AI as an
infrastructure. From the mines where the core components of hardware
originate, to the warehouses where human labourers are mere servants to the
automated structure, in an uneasy reminder of Chaplin’s Modern Times, to the
hardship of data classification by low paid workers in data labelling farms,
Crawford exposes the hard reality of the hidden side of AI success,
positioning that AI is not only not intelligent, it is also not artificial.
Concluding with the powerful reminder that AI is not an objective, neutral and
universal computational technique, Crawford reminds us that AI is deeply
embedded in the social, political, cultural and economic reality of those that
build, use and mostly control it [11].

###  2.3 AI is not the algorithm

AI is based on algorithms, but then so is any computer program and most of the
technologies around us. Nevertheless, the concept of ‘algorithm’ is achieving
magical proportions, used right and left to signify many things, de facto seen
as a synonym to AI.

The easiest way to understand an algorithm is as a recipe, a set of precise
rules to achieve a certain result. Every time you multiply two numbers, you
are using an algorithm, as well as you are when you are baking an apple pie.
However, by itself, the recipe has never turned into an apple pie; and, the
end result of your pie has as much to do with your baking skills and your
choice of ingredients, as with the choice for a specific recipe. The same
applies to AI algorithms: for a large part the behaviour and results of the
system depends on its input data, and on the choices made by those that
developed, trained and selected the algorithm. In the same way as we have the
choice to use organic apples to make our pie, in AI we also have the choice to
use data that respects and ensures fairness, privacy, transparency and all
other values we hold dear. This is what Responsible AI is about, and includes
demanding the same requirements from the ones that develop the systems that
affect us.

###  2.4 AI is a socio-technical system

If AI is not intelligent, nor magic, nor business as usual, nor an algorithm,
how best can we describe AI in order to take into account not only its
capabilities but also its societal implications?

AI is first and foremost technology that can automatise (simple, lesser) tasks
and decision making processes. At the present, AI systems are largely
incapable of understanding meaning. An AI system can correctly identify cats
in videos or cancer cells in scan images, but it has no idea of what a cat or
a cancer cell is. Moreover, AI system can only do this if there are enough
people performing the tasks (classification, collection, maintenance…) that
are needed to make the system function, misleadingly, in an autonomous manner.

At the same time, considering its societal impact and need for human
contribution, AI is much more than an automation technique. In this sense, AI
can best be understood as a socio-technical ecosystem, recognising the
interaction between people and technology, and how complex infrastructures
affect and are affected by society and by human behaviour. As such, AI is not
just about the automation of decisions and actions, the adaptability to learn
from the changes affected in the environment, and the interactivity required
to be sensitive to the actions and aims of other agents in that environment,
and decide when to cooperate or to compete. It is mostly about the structures
of power, participation and access to technology that determine who can
influence which decisions or actions are being automated, which data,
knowledge and resources are used to learn from, and how interactions between
those that decide and those that are impacted are defined and maintained.

A responsible, ethical, approach to AI will ensure transparency about how
adaptation is done, responsibility for the level of automation on which the
system is able to reason, and accountability for the results and the
principles that guide its interactions with others, most importantly with
people. In addition, and above all, a responsible approach to AI makes clear
that AI systems are artefacts manufactured by people for some purpose, and
that those which make these have the power to decide on the use of AI. It is
time to discuss how power structures determine AI and how AI establishes and
maintains power structures, and on the balance between, those who benefit
from, and those who are harmed by the use of AI [7].

##  3 Ensuring the responsible development and use of AI

Ethical AI is not, as some may claim, a way to give machines some kind of
‘responsibility’ for their actions and decisions, and in the process,
discharge people and organisations of their responsibility. On the contrary,
ethical AI gives the people and organisations involved more responsibility and
more accountability: for the decisions and actions of the AI applications, and
for their own decision of using AI in a given application context. When
considering effects and the governance thereof, the technology, or the
artefact that embeds that technology, cannot be separated from the socio-
technical ecosystem of which it is a component. Guidelines, principles and
strategies to ensure trust and responsibility in AI, must be directed towards
the socio-technical ecosystem in which AI is developed and used. It is not the
AI artefact or application that needs to be ethical, trustworthy, or
responsible. Rather, it is the social component of this ecosystem that can and
should take responsibility and act in consideration of an ethical framework
such that the overall system can be trusted by the society. Having said this,
governance can be achieved by several means, softer or harder. Currently
several directions are being explored, the main ones are highlighted in the
remainder of this section. Future research and experience will identify which
approaches are the most suitable, but given the complexity of the problem, it
is very likely that a combination of approaches will be needed.

###  3.1 Regulation

AI regulation is a hot topic, with many supporters and opponents. The recent
AI Act proposed by the European Commission envisions a risk-based approach to
regulation that ensures that people can trust that AI technology is used in a
way that is safe and compliant with the law, including the respect of
fundamental human rights333See https://eur-lex.europa.eu/legal-
content/EN/TXT/?uri=CELEX%3A52021PC0206.. The draft proposes to implement most
of the seven requirements put forward in the Ethics Guidelines for Trustworthy
AI as specific requirements for ‘high-risk’ AI applications. However, it does
not deal explicitly with issues of inclusion, non-discrimination and fairness.
Minimising or eliminating discriminatory bias or unfair outcomes is more than
excluding the use of low-quality data. The design of any artefact, such as an
AI system, is in itself an accumulation of choices and choices are biased by
nature as they involve selecting an option over another. Technical solutions
at dataset level must be complemented by socio-technical processes that help
avoid any discriminatory or unfair outcomes of AI.

Successful regulation demands clear choices about what is being regulated: the
AI Act draft does not yet clearly indicate if it is mostly focusing on AI
technology, or on the impact and results of its application? By defining AI
systems as using ‘machine learning, logic, or statistical approaches’ it opens
the door to endless semantic debates. Not only is the line between old-
fashioned computing and AI irrelevant, it is also blurry. What matters is
whether it poses risks to people, society and/or environment.

A future-proof regulation should focus on the outcomes of systems, whether or
not these systems fall in the current understanding of what is ‘AI’. If
someone is wrongly identified, is denied human rights or access to resources,
or is conditioned to believe or act in a certain way, it does not matter
whether the system is ‘AI’ or not. It is simply wrong. Moreover, regulation
must also address the inputs, processes and conditions under which AI is
developed and used are at least as important. Much has been said about the
dangers of biased data, and discriminating applications. Attention for the
societal, environmental and climate costs of AI systems is increasing. All
these must be included in any effort to ensure the responsible development and
use of AI.

At the same time, AI systems are computer applications, i.e. are artefacts,
and as such subject to existing constraints, legislation, for which due
diligence obligations and liabilities apply. That is, already now, AI does not
operate in a lawless space. Before defining extra regulations, we need to
start by understanding what is already covered by existing legislation.

A risk-based approach to regulation, as proposed by the European Commission,
is the right direction to take, but needs to be informed by a clear
understanding of what is the source of those risks. Moreover, it requires to
not merely focus on technical solutions at the level of the algorithms or the
datasets, but rather on developing socio-technical processes, and the
corporate responsibility, to ensure that any discriminatory or unfair outcomes
are avoided and mitigated. Independently of whether we call the system ‘AI’ or
not.

###  3.2 Standardisation

Standards are consensus-based agreed-upon ways of doing things by providing
what they consider to be the minimum universally-acknowledged specifications.
Industry standards are proven to be beneficial to organisations and
individuals. Standards can help reduce costs and improve efficiency of
organisations by providing consistency and quality metrics, the establishment
of a common vocabulary, good-design methodologies, and architectural
frameworks. At the same time, standards provide consumers with confidence in
the quality and safety of products and services. Most standards are considered
soft governance; i.e. non-mandatory to follow. Yet, it is often in the best
interest of companies to follow them to demonstrate due diligence and,
therefore, limit their legal liability in case of an incident. Moreover,
standards can ensure user-friendly integration between products [29]. AI
standards work to support the governance of AI development and use is ongoing
at ISO and IEEE, the two leading standards bodies. Such standards can support
AI policy goals in particular where it concerns safety, security and
robustness of AI, guarantees of explainability, and means to reduce bias in
algorithmic decisions [5]. Jointly with the International Electrotechnical
Commission (IEC), ISO has established a Standards Committee on Artificial
Intelligence (SC-42)444See https://www.iso.org/committee/6794475.html. but
ongoing SC-42 efforts are, so far, limited and preliminary [5].

At the same time, IEEE’s Standards Association global initiative on Ethically
Aligned Design555See https://ethicsinaction.ieee.org/. is actively working on
vision and recommendations to address the values and intentions as well as
legal and technical implementations of autonomous and intelligent systems to
prioritise human well-being [4]. This is the joint work of over 700
international researchers and practitioners. In particular, the P7000 series
aims to develop standards that will eventually serve to underpin and scaffold
future norms and standards within a new framework of ethical governance for
AI/AS design. Currently, the P7000 working groups are working on candidate
standard recommendations to address issues as diverse as system design,
transparency in autonomous systems, algorithmic bias, personal, children,
student, and employer data governance, nudging, or, the identification and
rating the trustworthiness of news sources. Notably, the efforts on assessment
of impact of autonomous and intelligent systems on human well-being is now
available as an IEEE standard [15].

###  3.3 Assessment

Responsible AI is more than the ticking of some ethical ‘boxes’ or the
development of some add-on features in AI systems. Nevertheless, developers
and users can benefit from support and concrete steps to understand the
relevant legal and ethical standards and considerations when making decisions
on the use of AI applications. Impact assessment tools provide a step-by-step
evaluation of the impact of systems, methods or tools on aspects such as
privacy, transparency, explanation, bias, or liability [28]. It is important
to realise that even though these approaches “can never map the entire
spectrum of opportunities, risks, and unintended consequences of AI systems,
they may identify preferable alternatives, valuable courses of action, likely
risks, and mitigating strategies. This has a dual advantage. As an opportunity
strategy, foresight methodologies can help leverage ethical solutions. As a
form of risk management, they can help prevent or mitigate costly mistakes, by
avoiding decisions or actions that are ethically unacceptable” [28].

Currently, much effort is being put on the development of assessment tools.
The EU Guidelines for trustworthy AI are accompanied by a comprehensive
assessment framework which was developed based on a public consultation
process. Finally, it is important to realise that any requirements for
trustworthy AI are necessary but not sufficient to develop human-centred AI.
That is, such requirements need to be understood and implemented from a
contextual perspective, i,e, it should be possible to adjust the
implementation of the requirement such as transparency based on the context in
which the system is used. I.e. requirements such as transparency should not
have one fixed definition for all AI systems, but rather be defined based on
how the AI system is used. At the same time, any AI technique used in the
design and implementation should be amenable to explicitly consider all
ethical requirements. E.g. it should be possible to explain (or to show) how
the system got to a certain decision or behaviour.

Assessment tools need to be able to account for this contextualisation, as
well as ensuring alignment with existing frameworks and requirements in terms
of other types of assessment, such that the evaluation of trust and
responsibility of AI systems provides added value to those developing and
using it, rather than adding yet another bureaucratic burden.

###  3.4 Codes of conduct and advisory boards

A professional code of conduct is a public statement developed for and by a
professional group to reflect shared principles about practice, conduct and
ethics of those exercising the profession; describe the quality of behaviour
that reflects the expectations of the profession and the community; provide a
clear statement to the society about these expectations, and enable
professionals to reflect on their own ethical decisions. A code of conduct
supports professionals to assess and resolve difficult professional and
ethical dilemmas. While there in the case of ethical dilemmas there is not a
correct solution, the professionals can give account of their actions by
referring to the code. In line with other socially sensitive professions, such
as medical doctors or lawyers, i.e., with the attendant certification of
‘ethical AI’ can support trust. Several organisations are working on the
development of codes of conduct for data and AI related professions, with
specific ethical duties. Just recently ACM, the Association for Computing
Machinery, the largest international association of computing professionals,
updated their code of conduct. This voluntary code is ‘‘a collection of
principles and guidelines designed to help computing professionals make
ethically responsible decisions in professional practice. It translates broad
ethical principles into concrete statements about professional
conduct666https://www.acm.org/code-of-ethics”. This code explicitly addresses
issues associated with the development of AI systems’, namely issues of
emergent properties, discrimination and privacy. Specifically, it calls out
the responsibility of technologists to ensure that systems are inclusive and
accessible to all, and requires that they are knowledgeable about privacy
issues.

Many organisations have since established the role of chief AI ethics officer,
or similar. Others, recognising that the societal and ethical issues that
arise from AI are complex and multi-dimensional, and therefore require
insights and expertise from many different disciplines and an open
participation of different stakeholders, have established AI ethics boards or
advisory panels. These roles and organisational components, are becoming a hot
topic as large businesses are increasingly dependent on AI and as the impact
of these systems on people and society becomes increasingly more evident, and
not always for the best. Recent scandals both about the impact of AI in bias
and discrimination, as on the way businesses are dealing with their own
responsibility, specifically on the role and treatment of whistle-blowers,
have increased the demand for clear and explicit organisational structures to
deal with the impact of AI.

###  3.5 Awareness and Participation

Inclusion and diversity are a broader societal challenge and central to AI
development. It is therefore important that as broad a group of people as
possible have a basic knowledge of AI, what can (and can’t) be done with AI,
and how AI impacts individual decisions and shapes society. A well-known
initiative in this area is Elements of AI777See
https://www.elementsofai.com/., initiated in Finland with the objective to
train one percent of EU citizens in the basics of artificial intelligence,
thereby strengthening digital leadership within the EU.

In parallel, research and development of AI systems must be informed by
diversity, in all the meanings of diversity, and obviously including gender,
cultural background, and ethnicity. Moreover, AI is not any longer an
engineering discipline and at the same time there is growing evidence that
cognitive diversity contributes to better decision making. Therefore,
developing teams should include social scientists, philosophers, and others,
as well as ensuring gender, ethnicity and cultural differences. It is equally
important to diversify the discipline background and expertise of those
working on AI to include AI professionals with knowledge of, amongst others,
philosophy, social science, law and economy. Regulation and codes of conduct
can specify targets and goals, along with incentives, as a way to foster
diversity in AI teams [10].

##  4 From a rational to a relational conception of AI

The dominant approach to AI has so far been an individualistic, rational one.
Russell and Norvig’s classic AI text book, defines AI along two dimensions
[26]: how it reasons (human-like or rationally ) and what it ‘does’ (think or
act). This classification does not mean that human reasoning is not rational,
but in their definition, human-like approaches aim to understand and model how
the human mind works, and rational approaches aim developing systems that that
result in the optimal level of benefit or utility for an individual. Both
approaches are well aligned with the central element of Western philosophy
“Cogito ergo sum” (I think therefore I am), and fundamentally conceptualise an
AI system as an individual entity capable of reasoning.

Throughout the whole history of AI, intelligent agents are characterised as
bounded rational, acting towards their own perceived interests [27]. For
instance, by identifying and applying patterns in (human-generated) data,
machine learning systems mimic and extend the human reasoning and actions
embedded in that data, whereas symbolic logic approaches (the so called ‘good
old-fashioned AI’, or GOFAI) aim to capture the laws of rational thought and
action, resulting in an idealised model of human reasoning. In both cases, the
view is that of an entity that reasons…

Rationality is often a central assumption for agent deliberation [26, 8].
Moreover, intelligent systems are expected to hold consistent world views
(beliefs), and to optimise action and decision based on a set of given
preferences (often accuracy has highest priority). This view on rationality
entails that agents are expected, and designed, to act rationally in the sense
that they choose the best means available to achieve a given end, and maintain
consistency between what is wanted and what is chosen [20].

The main advantages of a rationality assumption are their parsimony and
applicability to a very broad range of situations and environments, and their
ability to generate falsifiable, and sometimes empirically confirmed,
hypotheses about actions in these environments. This gives conventional
rational choice approaches a combination of generality and predictive power
not found in other approaches.

Both the human-like and the rational perspectives on AI described in [26] are
suited for a task-oriented view on the purpose of AI systems. That is, for
situations where the system is expected to optimise the result of its actions
for a specific purpose. Unfortunately, this type of reasoning mostly fits
those situations where all required information is available or can be
gathered at will. It does not really suit most human behaviour which is based
on split-second decisions, on habits, on social conventions and on power
structures. Moreover, even when the AI agent is able to perceive its
environment and adapt accordingly, it is mostly unaware of its own role in
that environment, and of the fact that its actions contribute to change, and
therefore unable to handle the result of its action.

Given the large impact of AI on society, a new modelling paradigm is needed
that is able to account for a feedback loop of decision–action that is
fundamentally grounded on context. When the aim of AI systems is to align with
societal behaviour and to develop systems that are able to interact with
people in social settings, such rationality view is not enough to model human
behaviour. That is, AI modelling needs to follow a social paradigm that can
account for the reality in which human behaviour is neither simple nor
rational, but derives from a complex mix of mental, physical, emotional and
social aspects. Realistic AI applications must therefore consider situations
in which not all alternatives, consequences, and event probabilities can be
foreseen. It is impossible to “rationally” optimise utility, as the utility
function is not completely known, neither are the optimisation criteria known.
This renders rational choice approaches unable to accurately model and predict
a wide range of human behaviours. Already in 2010, we described how different
types of variations and models cater for different applications, while no
generic model exists that serves as a foundation for all models [12].

##  5 Elements of a relational paradigm for AI

AI, like all technology, affects and changes our world, which in turn changes
us. New paradigms are needed that address collective understanding and the
effect of change in context and the feedback loop from change back to the
individual and collective reasoning and behaviour. Modelling this feedback
loop recognises that it is not just about which action is performed but, what
kind of reasoning leads to that action, and which values and perceptions lead
the observation of the context. Current AI paradigms strengthen power
structures and prevent a truly societal understanding of the impact and
challenges of AI for humanity and society. Theories and models to support the
understanding of how AI is experienced by individuals and social groups,
rather than how it is designed or operationalised, are mostly lacking.

Given the impact AI systems can have on people, inter-personal interactions,
and society as a whole, it seems to be relevant to consider a relational
stance to approach the specification, development and analysis of AI systems.
Both feminist theories and non-Western philosophies (in particular the Ubuntu
philosophy) take a societal rather than an individual stance, which may
provide a suitable grounds for a relational conceptualisation of AI. In this
section, I will reflect on possible ways on how Ubuntu and feminist
philosophies could serve this purpose.

###  5.1 Ubuntu philosophy and AI

Ubuntu expresses the deeply-held African ideals of one’s identity being rooted
in one’s interconnectedness with others, and emphasises norms for
interpersonal relationships that contribute to social justice, such as
reciprocity, selflessness and symbiosis. Community is at the core of Ubuntu,
focusing on interconnectedness and caring for communal living, underpinned by
values of cooperation and collaboration [23]. Solidarity, which requires
people to be aware of and attentive to the needs of those around them, rather
than focusing only on their own needs is therefore central in Ubuntu, with an
emphasis on caring, care-taking, and context [3]. As such, Ubuntu philosophy
is essentially relational, and defines morally right actions as those that
that connect, rather than separate (i.e. honours communal relationships,
reduces discord or promotes friendly relationships. The concept of community
can best be understood as an (objective) standard that should guide what the
majority wants, or what moral norms become central [16]. This does not imply
that individual rights are subordinated but that individuals pursue their own
good through pursuing the common good [21].

Human rights set the foundation of human dignity in terms of autonomy. This
view, for a large part originating from Kantian philosophy, sets human rights
as the ultimately ways of treating our intrinsically valuable capacity for
self-governance with respect. It has therefore been argued that the
collectivist grounds of Ubuntu thought are at odds with this individual
autonomy view. According to Metz, “While the Kantian theory is the view that
persons have a superlative worth because they have the capacity for autonomy,
the present, Ubuntu-inspired account is that they do because they have the
capacity to relate to others in a communal way.” [22, p.544]. Or, as Metz also
describes: “Human rights violations are ways of gravely disrespecting people’s
capacity for communal relationship, conceived as identity and solidarity […]”
[22, p.545]. In Ubuntu, human nature is special and inviolable due to its
capacity for harmonious relationships. At the same time, no individual’s
rights are greater than another, thus, every individual in a community,
including both children and adults is important and should be heard and
respected [24].

With respect to the ethics of AI development and use, the above formulation of
human dignity as the human capability to relate to others in a communal way,
can account for, or justify, the resolution of moral dilemmas, where autonomy
conflicts with beneficence or any of the other principles, as also proposed
with respect to bioethics and medicine, or to ground the UN’s sustainable
development goals[16].

Specifically, the formulation of Ubuntu described above may be used to justify
decisions in face of ethical dilemma(s), for example where such a decision
favours the action that enhances communal relationships, or the capacity for
the same [16]. As such, this framework could usefully supplement utilitarian,
individualistic and deontological approaches that are often embedded in AI
ethics decision-making. As proposed for the case of clinical contexts, ethical
decision making in the context of AI systems, can also be extended with rules
that state that “[a] breach of an ethical principle is justifiable if, on the
balance of probabilities, such a breach is more likely to enhance communal
relationships (…)” [16].

Given the transnational character of AI, it is imperative to address the ways
in which AI may impact or be accepted by society in various regions around the
world. In particular, it is needed to position the African continent in global
debates and policy-making in responsible AI. For instance, initiatives such as
Responsible AI Network – Africa , and the African Observatory on Responsible
AI are aiming at understand how AI may impact or be accepted by society in
various regions around the world, deepen the understanding of AI and its
effects in (Sub-Saharan) Africa, and promoting the development and
implementation of locally appropriate evidence-led AI policies and enabling
legislation.

It is as important to extend current conceptualisations of AI, with the
relational world view that characterise African thought, such as embedded in
Ubuntu philosophy. Current AI paradigms strengthen existing power structures
and prevent a truly societal understanding of the impact and challenges of AI
for humanity and society. Given the impact AI systems can have on people,
inter-personal interactions, and society as a whole, it seems to be relevant
to consider a relational stance to approach the specification, development and
analysis of AI systems. The deeply-held African vision of one’s identity being
rooted in connectedness to others and to society, as expressed in Ubuntu
philosophy, can support integrating such a social perspective to AI – in terms
of how it reasons (human-like or rationally) and what it ‘does’ (think or
act), which would result in a new paradigm that considers social/collective
reasoning and includes change, or reaction, as a third possible result of AI,
next to thinking and acting. Figure 1 depicts this perspective on AI,
extending the well-known dimensions defined in [26], here depicted in grey
shading.

###  5.2 Feminist theory and AI

As discussed in Section 2, the development, use and impact of AI is neither
‘artificial’ nor ‘intelligent’ but the product of choices involving theory and
values, strongly related by power structures that support and enforce an
increasing divide between the ‘haves’ and the ‘have-nots’. Current AI
paradigms, as seen in Section 4, rely, and are bound, by individualistic
theories of intelligence, thinking, rationality, and human nature but lack the
means to deal with values, social norms and collective reasoning, and how
these ground and influence human thinking and action. In social action,
outcomes are relative to the actions of others, where others are not just seen
as opponents or obstacles on the decision making, but as a positive force for
achieving a joint endeavour.

Already in 1995, Alison Adam proposed feminist epistemology as a radically
different, and qualitative, alternative to traditional views on the AI
narrative, centred on the question of whether we are prepared to accept and
use such systems in our lives, rather than reflecting on what constitutes
intelligence. A feminist approach analyses how AI systems may or may not
reinforce existing power structures, and how AI can or cannot represent
differing social groups [1]. More recently, Catherine D’Ignazio stressed this
need to think critically about the politics and ethics of representation [13].
Drawing a parallel with the role of cartographic maps as instruments of power,
producing worlds that are intimately bound up with that power, data analysis
and visualisation are also socially situated knowledge, ultimately determining
our interaction with and understanding of the AI system.

Many current approaches to ethics in AI approach decision in black and white,
good versus bad, under the idea that if only we can model ethics in some
formal, digital way, the problem would be solved, the AI system ‘be’ ethical,
and (other) people will be wiser and better human beings. What is usually
known as ‘Ethics by design’ has a very techno-solutionist ring to it… Reality
is messier; common-sense morality, moral indignation and assignment of guilt
and blame damage relationships, create separateness, and narrow our scope of
possible actions. Classic ethics theories are too abstract, practical
application leads to too many exceptions, and each can lead to a very
different ‘solution’ to any given dilemma.

Research on algorithmic injustice shows how AI and in particular Machine
Learning automates and perpetuates historical, often unjust and
discriminatory, patterns. The negative consequences of algorithmic systems,
especially on marginalized communities, require a deeper understanding of the
context in which AI systems are designed, developed and used, that needs go
much further than debiasing and/or making datasets inclusive [2]. Relational
ethics is borne out of a feminist ethics of care and recognises that objective
and rational moral reasoning cannot always advise on the ‘right’ way to
proceed – what is ‘right’ emerges within the dynamics of research
relationships and can depend on the time, place and people involved.
Relational ethics moves away from attempting to solve the ethical ‘problem’ to
asking the ethical ‘question’.

At the same time, a relational stance to AI also addresses collectivist views
and the continuous loop between change and decision. Modelling this feedback
loop recognises that the development and use of AI is not just about the
optimisation of performance but, most importantly, the societal and individual
context of the values and perceptions and reasoning that lead to action.
Addressing change from the human-like perspective, leads to approaches to AI
that aim at enhance, rather than replace, human performance, and from a
rational perspective, it concerns the development and optimisation of
institutional infrastructures that maximise the effects of rational behaviour.

All these perspectives need to be brought together to address the impact of AI
for a socially grounded and engaged perspective. This is no easy feat, but one
for which there are no single models, nor simple approaches. It will require
multidisciplinary and multi-stakeholder participation and to accept that any
solution is always contingent and contextual.

##  6 Conclusions

Increasingly, AI systems will be taking decisions that affect our lives, in
smaller or larger ways. In all areas of application, AI must be able to take
into account societal values, moral and ethical considerations, weigh the
respective priorities of values held by different stakeholders and in
multicultural contexts, explain its reasoning, and guarantee transparency. As
the capabilities for autonomous decision-making grow, perhaps the most
important issue to consider is the need to rethink responsibility. Being
fundamentally tools, AI systems are fully under the control and responsibility
of their owners or users. However, their potential autonomy and capability to
learn, require that design considers accountability, responsibility and
transparency principles in an explicit and systematic manner. The development
of AI algorithms has so far been led by the goal of improving performance,
leading to opaque black boxes. Putting human values at the core of AI systems
calls for a mind-shift of researchers and developers towards the goal of
improving transparency rather than performance, which will lead to novel and
exciting techniques and applications. In particular, this requires to
complement the currently predominant individualistic view of AI systems, to
one that acknowledges and incorporates the collective, societal,

Biological evolution has long been revised from a ‘ladder’ view: a uni-linear
progression from ‘primitive’ to ‘advanced’. The same revision is also seen in
anthropology: the idea that cultural evolution follows a ladder model, with
small-scale decentralised societies at the bottom and hierarchical, state,
societies at the top, where the top would be technologically more advanced,
has been shown to be not only demeaning but also inaccurate [14]. These fields
have long since moved to a more dynamic, branching type model that account to
interrelations between cooperation and competition, individualism and
collectivism, power and influence, cause and effect. Still, the main
perspective in AI, is that intelligence is linear, and by creating
increasingly more intelligent systems, we will solve the problems of previous
ones. It is high time that AI models embrace such a branching view. Only then,
can AI align with the diversity that truly reflects worldwide differences in
cultural and philosophical thought.

Rethinking AI from a relational, feminist, non-Western perspective is not a
fad or a thought experiment for philosophers. It is ultimately the only way
forward, for AI as a scientific field, and more importantly for all of us, and
for the world.

#### Acknowledgements

This work was partially supported by the Wallenberg AI, Autonomous Systems and
Software Program (WASP), funded by the Knut and Alice Wallenberg Foundation
and by the European Commission’s Horizon 2020 Research and Innovation
Programme project HumaneAI-Net (grant 825619).

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