Correction: A Convex Analysis Approach to Entropy Functions, Variational Principles and Equilibrium States

Correction to: Commun. Math. Phys. https://doi.org/10.1007/s00220-022-04403-z

1 Introduction

Let (X, d) be a metric space and ${\mathfrak {B}}$ its $\sigma$-algebra of Borel subsets of X. Denote by ${{\textbf{B}}}$ a Banach space over the field ${\mathbb {R}}$ equal to either

$$\begin{aligned} B_m(X)= & {} \big \{\varphi : \, X \,\rightarrow \, {\mathbb {R}} \,\,|\,\, \varphi \text { is Borel measurable and bounded}\big \}\\ \text { or } \,\,C_b(X)= & {} \big \{\varphi \in B_m(X) \,\,|\,\, \varphi \text { is continuous}\big \}\nonumber \\ \text { or else } \,\,C_c(X)= & {} \big \{\varphi \in C_b(X) \,\,|\,\, \varphi \text { has compact support}\big \} \nonumber \end{aligned}$$

endowed with the norm $\Vert \varphi \Vert _\infty = \sup _{x \, \in \, X}\, |\varphi (x)|$. In what follows, $\mathcal {P}_a(X)$ stands for the set of Borel finitely additive probability measures endowed with the total variation distance, $\mathcal {P}(X)$ denotes the set of Borel $\sigma$-additive probability measures on X with the weak$^*$ topology and C(X) is the space of real valued continuous maps whose domain is X.

Definition 1.1

A function $\Gamma :{{\textbf{B}}} \rightarrow {\mathbb {R}}$ is called a pressure function if it satisfies the following conditions:

($\hbox {C}_1$):

Monotonicity: $\varphi \leqslant \psi \,\, \Rightarrow \,\, \Gamma (\varphi ) \leqslant \Gamma (\psi ) \quad \forall \,\varphi , \, \psi \in {{\textbf{B}}}$.

($\hbox {C}_2$):

Translation invariance: $\Gamma (\varphi + c) = \Gamma (\varphi ) + c \quad \forall \,\varphi \in {{\textbf{B}}} \quad \forall \,c \in {\mathbb {R}}$.

($\hbox {C}_3$):

Convexity: $\Gamma (t \,\varphi + (1-t) \,\psi ) \leqslant t\, \Gamma (\varphi ) + (1-t)\, \Gamma (\psi ) \quad \forall \,\varphi , \, \psi \in {{\textbf{B}}} \quad \forall \, t \in \,[0,1]$.

The first result of [4] established the following abstract variational principle for pressure functions.

Theorem 1

[4] Let $\Gamma : {{\textbf{B}}} \rightarrow {\mathbb {R}}$ be a pressure function. Then

$$\begin{aligned} \Gamma (\varphi ) \,=\, \max _{\mu \, \in \, \mathcal {P}_a(X)}\, \left\{ {{\mathfrak {h}}}(\mu ) + \int \varphi \, d\mu \right\} \quad \quad \forall \, \varphi \in {{\textbf{B}}} \end{aligned}$$

(1.2)

where, for every $\mu \in \mathcal {P}_a(X)$,

$$\begin{aligned} {{\mathfrak {h}}}(\mu ) \,=\, \inf _{\varphi \, \in \, \mathcal {A}_\Gamma }\, \left\{ \int \varphi \, d\mu \right\} \quad \quad \text {and} \quad \quad \mathcal {A}_\Gamma \,=\, \big \{\varphi \in {{\textbf{B}}}:\, \Gamma (-\varphi ) \leqslant 0 \big \}. \end{aligned}$$

(1.3)

The map ${{\mathfrak {h}}}$ is concave, upper semi-continuous and

$$\begin{aligned} {{\mathfrak {h}}}(\mu ) \,=\, \inf _{\varphi \, \in \, {{\textbf{B}}}}\, \Big \{\Gamma (\varphi ) - \int \varphi \, d\mu \Big \} \quad \quad \forall \,\mu \in \mathcal {P}_a(X). \end{aligned}$$

Moreover, if $\alpha :\mathcal {P}_a(X) \rightarrow {\mathbb {R}} \cup \{-\infty \}$ is another function taking the role of ${{\mathfrak {h}}}$ in (1.2), then $\alpha \leqslant {{\mathfrak {h}}}$. If, in addition, X is locally compact and ${{\textbf{B}}} = C_c(X)$, then the maximum referred to in (1.2) is attained in ${\mathcal {P}}(X)$.

It is immediate from (1.2) that ${{\mathfrak {h}}}$ is upper bounded by $\Gamma (0)$. Since the pointwise infimum of concave functions is concave, and affine maps are themselves concave, we get from (1.3) that the map ${{\mathfrak {h}}}$ is concave. In [4, Theorem 1] we asserted incorrectly that ${{\mathfrak {h}}}$ is affine. The following example shows that this is not true in general.

Example 1.4

Consider a two point compact metric space $X=\{a,b\}$ with the discrete metric. The space ${\mathcal {P}}(X)$ of Borel probability measures on X is precisely the set $\big \{t\delta _a + (1-t)\delta _b\,:\, t \in [0,1]\big \}$ of convex combinations of the Dirac measures $\delta _a$ and $\delta _b$ supported on $\{a\}$ and $\{b\}$, respectively. Let $\Gamma :C(X) \rightarrow {\mathbb {R}}$ be given by

$$\begin{aligned} \varphi \in C(X) \quad \mapsto \quad \log \big (e^{\varphi (a)} + e^{\varphi (b)}\big ). \end{aligned}$$

This is a pressure function on the Banach space C(X) with the norm $\Vert \varphi \Vert _\infty = \max _{x \, \in \, X}\, |\varphi (x)|$. Actually, the formula for $\Gamma$ gives the topological pressure of the potentials on the shift space $\{0,1\}^{\mathbb {N}}$ that are locally constant on cylinders of size 1 (cf. [6]). Thus, Theorem 1 provides the variational principle

$$\begin{aligned} \Gamma (\varphi ) \,=\, \max _{\mu \,\in \, \mathcal {P}(X)}\, \Big \{{\mathfrak {h}}(\mu ) + \int \varphi \, d\mu \Big \} \end{aligned}$$

where

$$\begin{aligned} {\mathfrak {h}}(\mu ) \, = \, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \int \psi \,d\mu \quad \quad \forall \, \mu \in \mathcal {P}(X) \end{aligned}$$

(1.5)

and

$$\begin{aligned} \mathcal {A}_\Gamma \, = \, \Big \{\psi \in C(X) :\,\Gamma (-\psi ) \leqslant 0\Big \} \, = \, \Big \{\psi \in C(X) :\, e^{-\psi (a)} + e^{-\psi (b)} \,\leqslant \, 1\Big \}.\qquad \end{aligned}$$

(1.6)

Therefore, there is $\mu _0 \in \mathcal {P}(X)$ such that

$$\begin{aligned} \Gamma (0) \,=\, \log 2 \, = \,{\mathfrak {h}}(\mu _0). \end{aligned}$$

Yet, a straightforward computation using (1.5) and (1.6) yields

$$\begin{aligned} {\mathfrak {h}}(\delta _a) \,=\, \, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \, \psi (a) \,=\, 0 \quad \quad \text { and } \quad \quad {\mathfrak {h}}(\delta _b) \,=\, \, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \, \psi (b) \,=\, 0. \end{aligned}$$

Thus, ${\mathfrak {h}}$ is not affine.

2 Necessary and Sufficient Conditions for the Affinity of ${\mathfrak {h}}$

The wrong statement that ${\mathfrak {h}}$ is always affine is the content of Lemma 3.2 in [4]. Although this error has no impact on the other results of [4], since the possible affinity of ${\mathfrak {h}}$ is not needed elsewhere in that paper, it is relevant to identify where the proof of Lemma 3.2 fails and to clarify under what assumptions it becomes correct.

Let X be a locally compact metric space. The convex set $\mathcal {P}(X)$ is compact and metrizable with the weak$^*$ topology, as a consequence of Banach–Alaoglu Theorem (cf. [7, Theorem 2, V.4.2]). So, by the Krein–Milman Theorem (cf. [8, pages 187–188]), it is the closed convex hull of its extreme points. Moreover, we can use the Choquet Representation Theorem (cf. [3, Theorem 6.6]) to express each member of $\mathcal {P}(X)$ as a generalized convex combination of the extreme elements of $\mathcal {P}(X)$. More precisely, if E(X) denotes the set of extreme points of $\mathcal {P}(X)$ and $\mu$ belongs to $\mathcal {P}(X)$, then there is a unique measure ${\mathbb {P}}_\mu$ on the Borel $\sigma$-algebra of $\mathcal {P}(X)$ such that ${\mathbb {P}}_\mu (E(X)) = 1$ and $\mu = \int _{E(X)}\,m\, \,d{\mathbb {P}}_\mu (m)$. We call the last equality the decomposition in extremes of $\mu$.

Given a pressure function $\Gamma$, take the map ${\mathfrak {h}}$ provided by Theorem 1 and consider $\mu \in \mathcal {P}(X)$. In the Banach space ${{\textbf{B}}}$, define the following binary relation

$$\begin{aligned} \varphi \,\preceq _{\mu }\, \psi \quad \Leftrightarrow \quad \int \varphi \,d\mu \,\leqslant \, \int \psi \, d\mu . \end{aligned}$$

Then $\preceq _{\mu }$ is well defined, total (since $({\mathbb {R}}, \leqslant )$ is completely ordered), reflexive and transitive, but depends on the fixed probability measure $\mu$.

Definition 2.1

A non-empty set A with a pre-order $\lhd$ is said to be downward directed if, given $a, b \in A$, there is $c \in A$ such that $c \lhd a$ and $c \lhd b$.

Lemma 2.2

Given $\mu \in \mathcal {P}(X)$, if ${\mathfrak {h}}(\mu ) > -\infty$ then $\mathcal {A}_\Gamma$ with the pre-order $\preceq _{\mu }$ is a lower bounded downward directed set in ${{\textbf{B}}}$.

Proof

Given $\varphi , \,\psi \in \mathcal {A}_\Gamma$, one has either $\int \varphi \,d\mu \,\leqslant \, \int \psi \, d\mu$ or $\int \psi \,d\mu \,\leqslant \, \int \varphi \, d\mu$. In the former case, the map $\Psi = \varphi$ is in $\mathcal {A}_\Gamma$ and satisfies both $\Psi \preceq _{\mu } \varphi$ and $\Psi \preceq _{\mu } \psi$. In the latter case, one chooses $\Psi = \psi$. Thus, $(\mathcal {A}_\Gamma , \preceq _{\mu })$ is a downward directed set.

As ${\mathfrak {h}}(\mu ) > -\infty$, we may take the constant map $H:X \rightarrow {\mathbb {R}}$ defined by $H(x) = {\mathfrak {h}}(\mu )$, which is a lower bound of $(\mathcal {A}_\Gamma , \preceq _{\mu })$ in ${{\textbf{B}}}$ since

$$\begin{aligned} \int H(x) \,d\mu (x) = \int \,{\mathfrak {h}}(\mu ) \, d\mu (x) = {\mathfrak {h}}(\mu ) = \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \int \psi \,d\mu \,\leqslant \, \int \psi \, d\mu \quad \quad \forall \psi \in \mathcal {A}_\Gamma . \end{aligned}$$

$\square$

Therefore, $\big (\int \psi \, d\mu \big )_{\psi \, \in \, \mathcal {A}_\Gamma }$ is a lower bounded decreasing net. So, as X is locally compact, if ${{\textbf{B}}} = C_c(X)$ and $\mu \in \mathcal {P}(X)$, then we may apply the Monotone Convergence Theorem for nets (cf. [9, Theorem IV.15] or [5, Theorem 1, Chapter IV, §1]), thus concluding that the infimum of the net with respect to $\preceq _{\mu }$, say ${\inf }^{(\mu )} \mathcal {A}_\Gamma$, belongs to $L^1(X, {\mathfrak {B}}, \mu )$ and

$$\begin{aligned} \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \,\int \psi \, d\mu \, = \,\int {\inf }^{(\mu )} \mathcal {A}_\Gamma \,d\mu . \end{aligned}$$

Proposition 2.3

Assume that X is locally compact and $\inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) > - \infty$. Then ${\mathfrak {h}}$ is affine if and only if

$$\begin{aligned} \int {\inf }^{(\mu )} \mathcal {A}_\Gamma \,d\mu = \int _{E(X)}\,\left( \int {\inf }^{(m)} \, \mathcal {A}_\Gamma \, (x) \, dm(x) \right) \,d{\mathbb {P}}_\mu (m) \quad \forall \, \mu \in \mathcal {P}(X)\qquad \end{aligned}$$

(2.4)

where $\mu = \int _{E(X)}\,m\, \,d{\mathbb {P}}_\mu (m)$ is the decomposition in extremes of $\mu$.

Proof

Assume that $\inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) > - \infty$ and the condition (2.4) is valid. Then, for every $\mu \in \mathcal {P}(X)$,

$$\begin{aligned} {\mathfrak {h}}(\mu )= & {} \,\inf _{\psi \, \in \, \mathcal {A}_\Gamma } \,\int \psi \, d\mu \,=\, \,\int {\inf }^{(\mu )} \mathcal {A}_\Gamma \,d\mu \nonumber \\= & {} \,\int _{E(X)}\,\left( \int {\inf }^{(m)} \, \mathcal {A}_\Gamma \, (x) \, dm(x) \right) \,d{\mathbb {P}}_\mu (m) \nonumber \\= & {} \,\int _{E(X)}\,\left( \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \, \int \psi (x) \, dm(x)\right) \,d{\mathbb {P}}_\mu (m) \nonumber \\= & {} \int _{E(X)}\,{\mathfrak {h}}(m)\,d{\mathbb {P}}_\mu (m). \end{aligned}$$

(2.5)

Remark 2.6

We note that, to obtain the previous estimates, which yield the equality

$$\begin{aligned} {\mathfrak {h}}(\mu ) \,= \, \int _{E(X)}\,{\mathfrak {h}}(m)\,d{\mathbb {P}}_\mu (m) \quad \quad \forall \, \mu \in \mathcal {P}(X) \end{aligned}$$

(2.7)

we only needed to assume that ${\mathfrak {h}}(\mu ) > -\infty$ for every $\mu \in \mathcal {P}(X)$.

We are left to conclude from (2.5) that ${\mathfrak {h}}$ is affine. Given $\mu _1,\,\mu _2 \in \mathcal {P}(X)$ whose decompositions in extremes are $\mu _i = \int _{E(X)}\,m\,\,d{\mathbb {P}}_{\mu _i}(m)$, for $i \in \{1,2\}$, then

$$\begin{aligned} \forall \, 0< t < 1 \quad \quad t \,\mu _1 + (1-t)\, \mu _2 \,=\, \int _{E(X)}\,m \, \big [t \,d{\mathbb {P}}_{\mu _1} + (1-t) \,d{\mathbb {P}}_{\mu _2}\big ](m) \end{aligned}$$

and so, using (2.5), we get for every $0< t < 1$

$$\begin{aligned} {\mathfrak {h}}\big (t\, \mu _1 + (1-t) \,\mu _2\big ) \, = \,\int _{E(X)} {\mathfrak {h}}(m) \,\,d[t\, {\mathbb {P}}_{\mu _1} + (1-t)\,{\mathbb {P}}_{\mu _2}](m) \,=\, t\, {\mathfrak {h}}(\mu _1) + (1-t)\, {\mathfrak {h}}(\mu _2).\nonumber \\ \end{aligned}$$

(2.8)

Remark 2.9

It may happen that there is $\nu _0 \in \mathcal {P}(X)$ such that ${\mathfrak {h}}(\nu _0) = -\infty$, so the first assumption in Proposition 2.3 fails, whereas the map ${\mathfrak {h}}$ is affine. This is precisely the case illustrated in Example 2.14, for which, despite its infinite values, the map ${\mathfrak {h}}$ is affine. Actually, in this example the equality (2.7) is valid, and this equality is just what one uses in the computation (2.8) to show that ${\mathfrak {h}}$ is affine.

Remark 2.10

If we may find a pre-order so that the function $\inf \mathcal {A}_\Gamma$ does not depend on the probability measure $\mu$, then the request (2.4) in the statement of Proposition 2.3 is trivially fulfilled. See, for instance, Example 2.15, for which the set $\mathcal {A}_\Gamma$ is downward directed and lower bounded in C(X) with respect to the usual order, defined by $\varphi \,\leqslant \, \psi \Leftrightarrow \varphi (x) \,\leqslant \, \psi (x) \, \forall x \in X.$

Regarding the converse assertion in Proposition 2.3, assume now that $\inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) > - \infty$ and ${\mathfrak {h}}$ is affine. We start by recalling that:

Lemma 2.11

[10, page 186] If $F :\mathcal {P}(X) \rightarrow {\mathbb {R}}$ is affine, upper semi-continuous and lower bounded by a continuous map $G:\mathcal {P}(X) \rightarrow {\mathbb {R}}$, then

$$\begin{aligned} F(\mu ) \, = \, \int _{E(X)}\,F(m)\,d{\mathbb {P}}_\mu (m) \quad \quad \forall \, \mu \in \mathcal {P}(X) \end{aligned}$$

(2.12)

where $\mu = \int _{E(X)}\,m\, \,d{\mathbb {P}}_\mu (m)$ is the decomposition in extremes of $\mu$.

Proof

Firstly, suppose that F is continuous. As F is affine, then the equality (2.12) is an immediate consequence of the Choquet Representation Theorem.

Suppose now that F is only upper semi-continuous. Then F is the pointwise limit of a decreasing (with respect to the usual order in C(X)) net $(F_\alpha )_\alpha$ of continuous affine maps $F_\alpha :\mathcal {P}(X) \rightarrow {\mathbb {R}}$ (cf. [2, Corollary I.1.4]). As $G \leqslant F = \inf _{\alpha } \,F_\alpha$, then $(F_\alpha )_\alpha$ is bounded from below in the space of continuous real valued maps whose domain is $\mathcal {P}(X)$. Therefore, for every $\mu \in \mathcal {P}(X)$, whose decomposition in extremes is $\mu = \int _{E(X)}\,m\, \,d{\mathbb {P}}_\mu (m)$, one has

$$\begin{aligned} F(\mu )= & {} \lim _{\alpha } F_\alpha (\mu ) = \lim _{\alpha } \int _{E(X)} F_\alpha (m) \, d{\mathbb {P}}_\mu (m) \\= & {} \int _{E(X)} \lim _{\alpha } \, F_\alpha (m) \, d{\mathbb {P}}_\mu (m) \,=\, \int _{E(X)} F(m) \, d{\mathbb {P}}_\mu (m) \end{aligned}$$

where the last but one equality is due to the Monotone Convergence Lemma for nets (cf. [1, Lemma 19.36]), which may be applied since ${\mathbb {P}}_\mu$ is $\sigma$-additive on the Borel $\sigma$-algebra of the compact metric space $\mathcal {P}(X)$. $\square$

Since ${\mathfrak {h}}$ is upper semi-continuous and satisfies $\inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) > - \infty$, then, if in addition it is affine, one may apply Lemma 2.11 using $F={\mathfrak {h}}$ and $G = \inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu )$. Therefore,

$$\begin{aligned} {\mathfrak {h}}(\mu ) \, = \, \int _{E(X)}\,{\mathfrak {h}}(m)\,d{\mathbb {P}}_\mu (m) \quad \quad \forall \, \mu \in \mathcal {P}(X). \end{aligned}$$

That is,

$$\begin{aligned} \int {\inf }^{(\mu )} \mathcal {A}_\Gamma \,d\mu \, = \, \int _{E(X)}\,\left( \int {\inf }^{(m)} \, \mathcal {A}_\Gamma \, (x) \, dm(x) \right) \,d{\mathbb {P}}_\mu (m) \quad \quad \forall \, \mu \in \mathcal {P}(X). \end{aligned}$$

The proof of Proposition 2.3 is completed. $\square$

Remark 2.13

In Example 1.4, one has $\inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) > - \infty$. Indeed, ${\mathfrak {h}}(\delta _a) = {\mathfrak {h}}(\delta _b) = 0$ and

$$\begin{aligned} {\mathfrak {h}}\big (t \,\delta _a + (1-t) \,\delta _b\big ) \, = \, -t \log \,(t) - (1-t) \log \,(1-t)\quad \quad \forall \, 0< t < 1. \end{aligned}$$

So $\inf _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) = \min _{\mu \, \in \, \mathcal {P}(X)} \, {\mathfrak {h}}(\mu ) = 0$. However, if $\mu _0 = \frac{1}{2}\, \delta _a + \frac{1}{2}\, \delta _b$, then

$$\begin{aligned} \log 2 \, = \, \int {\inf }^{(\mu _0)} \mathcal {A}_\Gamma \,d\mu _0 \quad \ne \quad \int _{E(X)\,=\,\{\delta _a, \, \delta _b\}}\,\left( \int {\inf }^{(m)} \, \mathcal {A}_\Gamma \, (x) \, dm(x) \right) \,d{\mathbb {P}}_{\mu _0}(m) \,=\, 0. \end{aligned}$$

Example 2.14

Consider, as in Example 1.4, the set $X = \{a,b\}$ with the discrete metric and its space of Borel probability measures $\mathcal {P}(X) = \big \{t\delta _a + (1-t)\delta _b \,:\, t \in [0,1]\big \}$. Let now $\Gamma :C(X) \rightarrow {\mathbb {R}}$ be given by

$$\begin{aligned} \Gamma (\varphi ) \,=\, \varphi (a). \end{aligned}$$

It is easy to check that $\Gamma$ is a pressure function on the Banach space C(X). Therefore, by Theorem 1,

$$\begin{aligned} \Gamma (\varphi ) \,=\, \max _{\mu \,\in \,\mathcal {P}(X)}\, \Big \{{\mathfrak {h}}(\mu ) + \int \varphi \, d\mu \Big \} \end{aligned}$$

where

$$\begin{aligned} {\mathfrak {h}}(\mu ) \, = \, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \int \psi \,d\mu \quad \quad \forall \, \mu \in \mathcal {P}(X) \end{aligned}$$

and

$$\begin{aligned} \mathcal {A}_\Gamma \,= & {} \, \Big \{\psi \in C(X) :\,\Gamma (-\psi ) \leqslant 0\Big \} \, = \, \Big \{\psi \in C(X) :\, -\psi (a) \,\leqslant \, 0\Big \} \, \\= & {} \, \Big \{\psi \in C(X) :\, \psi (a) \,\geqslant \, 0\Big \}. \end{aligned}$$

Thus,

$$\begin{aligned} {\mathfrak {h}}(\delta _a)= & {} \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \int \psi \,d\delta _a \,=\, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \, \psi (a) \,=\, \inf _{\{\psi :\,\psi (a) \, \geqslant \,0\}} \, \psi (a)\,=\, 0; \\ {\mathfrak {h}}(\delta _b)= & {} \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \int \psi \,d\delta _b \,=\, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \, \psi (b) \,=\, \inf _{\{\psi :\,\psi (a) \, \geqslant \,0\}} \, \psi (b)\,=\, -\infty ; \end{aligned}$$

and, for every $0< t < 1$,

$$\begin{aligned} {\mathfrak {h}}\big (t\,\delta _a + (1-t)\,\delta _b\big )= & {} \, \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \int \psi \,d(t\,\delta _a + (1-t)\,\delta _b) \\= & {} \inf _{\psi \, \in \, \mathcal {A}_\Gamma } \, \big (t\,\psi (a) + (1-t)\,\psi (b)\big ) \\= & {} \inf _{\{\psi :\,\psi (a) \, \geqslant \,0\}} \, \big (t\,\psi (a) + (1-t)\,\psi (b)\big )\\= & {} -\infty . \end{aligned}$$

Thus, ${\mathfrak {h}}$ is affine, although for some probability measures one has ${\mathfrak {h}} = -\infty$.

Apart from the known cases where ${\mathfrak {h}}$ is affine when restricted to an interesting convex subset of $\mathcal {P}(X)$ (see, for instance, [4, Remark 2.4]), there are examples for which a better choice of the pre-order in $\mathcal {A}_\Gamma$ is available.

Example 2.15

Consider a compact metric space (X, d) and the function $\Gamma :C(X) \rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \Gamma (\varphi ) \,=\, \max _{x \, \in \, X} \, \varphi (x). \end{aligned}$$

Then $\Gamma$ is a pressure function, since

• 1. $\varphi \,\leqslant \, \psi \,\, \Rightarrow \,\, \max _{x \, \in \, X} \, \varphi (x) \,\leqslant \, \max _{x \, \in \, X} \, \psi (x)$;

• 2. $\max _{x \, \in \, X} \, (\varphi + c)(x) \,=\, \big (\max _{x \, \in \, X} \, \varphi (x)\big ) + c \quad \forall \,\varphi \in C(X) \quad \forall \,c \in {\mathbb {R}}$;

• 3. $\forall \,\varphi , \, \psi \in C(X) \quad \forall \, t \in \,[0,1]$,

  $$\begin{aligned} \max _{x \, \in \, X} \, \big (t \,\varphi + (1-t) \,\psi \big )(x) \,\leqslant \, t\, \max _{x \, \in \, X} \, \varphi (x) + (1-t)\, \max _{x \, \in \, X} \, \psi (x). \end{aligned}$$

Moreover,

$$\begin{aligned} \mathcal {A}_\Gamma= & {} \Big \{\psi \in C(X) \,: \, \Gamma (-\psi ) \leqslant 0 \Big \} \,=\, \Big \{\psi \in C(X) \,: \, \max _{x \, \in \, X} \, -\psi (x) \leqslant 0 \Big \} \\= & {} \Big \{\psi \in C(X) \,: \, \psi \geqslant 0 \Big \}. \end{aligned}$$

So

$$\begin{aligned} {{\mathfrak {h}}}(\mu ) \,=\, \inf \Big \{\int \psi \,d \mu :\, \psi \in C(X) \text { and } \psi \geqslant 0\Big \} \,=\, 0 \quad \quad \forall \, \mu \in \mathcal {P}(X). \end{aligned}$$

Hence ${\mathfrak {h}}$ is affine. Furthermore, given $\varphi \in C(X)$ whose maximum is attained at a point $x_0 \in X$, one has

$$\begin{aligned} \Gamma (\varphi ) \, = \, \varphi (x_0) \, = \, {\mathfrak {h}}(\delta _{x_0}) + \int \varphi \, d\delta _{x_0}. \end{aligned}$$

In this case, instead of the pre-order $\preceq _{\mu }$, we may consider in $\mathcal {A}_\Gamma$ the usual order in C(X) and still conclude that $(\mathcal {A}_\Gamma , \leqslant )$ is a lower bounded downward directed set. Indeed, given $\varphi , \, \psi \in \mathcal {A}_\Gamma$, the map $\min \,\{\varphi , \, \psi \}$ is continuous and non-negative, hence belongs to $\mathcal {A}_\Gamma$; besides, one has $\min \,\{\varphi , \, \psi \} \leqslant \varphi$ and $\min \,\{\varphi , \, \psi \} \leqslant \psi$; and the map 0 is a lower bound for $\mathcal {A}_\Gamma$. Therefore, the function $\inf \mathcal {A}_\Gamma$ does not depend on any probability measure $\mu$.