Potentials.
Let \((J,{\mathcal {A}})\) be a measurable space. A measurable function \(F:~J\times J\rightarrow {\mathbb {R}}\) is a potential, if there is a measurable function \(f:~ J\rightarrow {\mathbb {R}}\) such that \(F(x,y)=f(x)-f(y)\). It is easy to see that a bounded measurable function \(F:~J\times J\rightarrow {\mathbb {R}}\) is a potential if and only if \(F(x,y)+F(y,z)+F(z,x)=0\) for all \(x,y,z\in J\).
Of particular importance will be cut potentials of the form \({\mathbb {1}}_A(x)-{\mathbb {1}}_A(y)={\mathbb {1}}_{A\times A^c}(x,y)-{\mathbb {1}}_{A^c\times A}(x,y)\), where \(A\in {\mathcal {A}}\). Every potential F can be expressed by cut potentials as
$$\begin{aligned} F(x,y) = \int \limits _{-C}^C ({\mathbb {1}}_{A_t}(x)-{\mathbb {1}}_{A_t}(y))\,dt, \end{aligned}$$
(9)
where C is an upper bound on |F|, and \(A_t\) (\(-C\le t\le C\)) is a measurable subset of J such that \(A_t\subseteq A_s\) for \(t<s\), \(\cap _t A_t=\emptyset \) and \(\cup _t A_t=J\). To see this, let \(F(x,y)=f(x)-f(y)\) for some bounded measurable function f, and define \(A_t=\{x\in J:~f(x)\ge t\}\) \((-C\le t\le C)\).
Circulations.
Circulations and potentials.
Recall that \(\alpha \in {\mathfrak {M}}({\mathcal {A}}^2)\) is a circulation if its two marginals \(\alpha ^1\) and \(\alpha ^2\) are equal. This is clearly equivalent to saying that
$$\begin{aligned} \alpha (X\times X^c)=\alpha (X^c\times X)\qquad (\forall X\in {\mathcal {A}}) \end{aligned}$$
(10)
(just cancel the common part \(X\times X\) in \(\alpha (X\times J)=\alpha (J\times X)\)). Circulations form a linear subspace \({\mathfrak {C}}={\mathfrak {C}}({\mathcal {A}})\) of the space \({\mathfrak {M}}({\mathcal {A}}^2)\) of finite signed measures.
In the finite case, circulations of the form \(\delta _{x_1x_2}+\dots +\delta _{x_{n-1}x_n}+\delta _{x_nx_1}\) generate the space of all circulations (even those with \(n\le 3\) do). In the measure case, this is not always so, as the next example shows.
Example 4.1
(Cyclic graphing and digraphing). For a fixed \(a\in (0,1)\), let \({\mathbf {C}}_a\) be the graphing on [0, 1] obtained by connecting every point x to \(x+a\pmod 1\) and \(x-a\pmod 1\). If a is irrational, this graph consists of two-way infinite paths; if a is rational, the graph will consist of cycles. We will also use the directed version \(\overrightarrow{C}_a\), obtained by connecting x to \(x+a\pmod 1\) by a directed edge.
The uniform measure \(\mu \) on the edges of \(\overrightarrow{C}_a\) is trivially a circulation, both of its marginals being the uniform measure \(\lambda \) on [0, 1). Every circulation \(\alpha \) supported on the edges is a constant multiple of this. Indeed, \(\alpha ^1(A)=\alpha (A\times (A+a))=\alpha ^2(A+a)=\alpha ^1(A+a)\) for every Borel set \(A\subseteq [0,1)\), which means that \(\alpha ^1\) is invariant under translation by a. It is well-known that only scalar multiples of \(\lambda \) have this property.
We need two lemmas describing “duality” relations between potentials and circulations.
Lemma 4.2
A signed measure \(\alpha \in {\mathfrak {M}}({\mathcal {A}}^2)\) is a circulation if and only if \(\alpha (F)=0\) for every potential F.
Proof
The “if” part follows by applying the condition to the potential \({\mathbb {1}}_{A}(x)-{\mathbb {1}}_{A}(y)\):
$$\begin{aligned} \alpha (A\times J)-\alpha (J\times A)= \int \limits _{J\times J}({\mathbb {1}}_{A}(x)-{\mathbb {1}}_{A}(y))\,d\alpha (x,y) =0. \end{aligned}$$
To prove the converse, let \(\alpha \) be a circulation, then for every potential \(F(x,y)=f(x)-f(y)\), we have
$$\begin{aligned} \alpha (F)=\int \limits _{J\times J}f(x)-f(y)\,d\alpha (x,y) = \int \limits _J f(x)\,d\alpha ^2(x)-\int \limits _J f(y)\,d\alpha ^1(y)=0.[-3.8pc] \end{aligned}$$
\(\square \)
Lemma 4.3
Let \({\mathcal {L}}:~{\mathfrak {M}}({\mathcal {A}}^2)\rightarrow {\mathbb {R}}\) be a continuous linear functional. Then \({\mathcal {L}}\) vanishes on the space \({\mathfrak {C}}\) of circulations if and only if there is a continuous linear functional \({\mathcal {K}}:~{\mathfrak {M}}({\mathcal {A}})\rightarrow {\mathbb {R}}\) such that \({\mathcal {L}}(\mu )={\mathcal {K}}(\mu ^1-\mu ^2)\) for all \(\mu \in {\mathfrak {M}}({\mathcal {A}}^2)\).
Proof
The kernel of the linear operator \(\varphi \mapsto \varphi ^1-\varphi ^2\) (\(\varphi \in {\mathfrak {M}}({\mathcal {A}}^2)\)) is \({\mathfrak {C}}\). The range of this operator is
$$\begin{aligned} \mathrm{Rng}({\mathcal {T}})=\{\nu \in {\mathfrak {M}}({\mathcal {A}}):~\nu (J)=0\}. \end{aligned}$$
(11)
Indeed, if \(\nu =\mu ^1-\mu ^2\in \mathrm{Rng}({\mathcal {T}})\), then \(\nu (J)=\mu (J\times J)-\mu (J\times J)=0\). Conversely, if \(\nu (J)=0\), then for any probability measure \(\gamma \) on \({\mathcal {A}}\),
$$\begin{aligned} {\mathcal {T}}(\gamma \times \nu ) = \gamma (J)\nu -\nu (J)\gamma = \nu , \end{aligned}$$
so \(\nu \) is in the range of \({\mathcal {T}}\). It is easy to check that \(\nu (J)=0\) defines a closed subspace of \({\mathfrak {M}}({\mathcal {A}})\). Hence Proposition 3.5 implies the necessity of the condition. The sufficiency is straightforward, since \(\mu ^1-\mu ^2=0\) for every circulation \(\mu \). \(\square \)
Let \({\mathcal {L}}\in {\mathfrak {C}}^\perp \) and \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\). Restricting \({\mathcal {L}}\) to measures \(\mu \ll \psi \), we get a more explicit representation: there is a potential F such that
$$\begin{aligned} {\mathcal {L}}(\mu )=\mu (F)\qquad (\mu \ll \psi ). \end{aligned}$$
(12)
Indeed, consider the continuous linear functional \({\mathcal {K}}\) constructed in Lemma 4.3, and its representation \({\mathcal {K}}(\nu )=\nu (g)\) by a bounded measurable function \(g:~J\rightarrow {\mathbb {R}}\) in Proposition 3.6, valid for every \(\nu \ll \psi ^1+\psi ^2\). Then for the potential \(F(x,y)=g(x)-g(y)\) and every \(\mu \ll \psi \),
$$\begin{aligned} \mu (F)&=\int \limits _{J\times J} g(x)-g(y)\,d\mu (x,y) = \mu ^1(g)-\mu ^2(g) = {\mathcal {K}}(\mu ^1-\mu ^2)={\mathcal {L}}(\mu ). \end{aligned}$$
Existence of circulations.
Now we begin to carry out our program of extending basic flow-theoretic results in combinatorial optimization to measures. Our first goal is to generalize the Hoffman Circulation Theorem and to characterize optimal circulations.
Given two measures \(\varphi \) and \(\psi \) on \(J\times J\), we can ask whether there exists a circulation \(\alpha \) such that \(\varphi \le \alpha \le \psi \). Clearly \(\varphi \le \psi \) is a necessary condition, but it is not sufficient in general. The following theorem generalizes the Hoffman Circulation Theorem.
Theorem 4.4
For two signed measures \(\varphi ,\psi \in {\mathfrak {M}}(J\times J)\), there exists a circulation \(\alpha \) such that \(\varphi \le \alpha \le \psi \) if and only if \(\varphi \le \psi \) and \(\varphi (X\times X^c)\le \psi (X^c\times X)\) for every set \(X\in {\mathcal {A}}\).
Proof
The necessity of the condition is trivial: if the circulation \(\alpha \) exists, then \(\varphi (X\times X^c)\le \alpha (X\times X^c)=\alpha (X^c\times X) \le \psi (X^c\times X)\).
To prove sufficiency, consider the set \({\mathfrak {X}}=\{\mu \in {\mathfrak {M}}({\mathcal {A}}^2):~\varphi \le \mu \le \psi \}\). We may assume (by adding a sufficiently large circulation, say \(|\varphi |+|\varphi |^*\)) that \(0\le \varphi \le \psi \). We want to prove that \({\mathfrak {C}}\cap {\mathfrak {X}}\not =\emptyset \).
First, we prove the weaker fact that
$$\begin{aligned} d_{\mathrm{tv}}({\mathfrak {C}},{\mathfrak {X}})=0. \end{aligned}$$
(13)
Suppose that \(c=d_{\mathrm{tv}}({\mathfrak {C}},{\mathfrak {X}})>0\). Let \({\mathfrak {X}}'=\{\mu \in {\mathfrak {M}}({\mathcal {A}}^2): ~d_\mathrm{tv}(\mu ,{\mathfrak {X}})<c\}\), then \({\mathfrak {X}}'\) is a convex open subset of \({\mathfrak {M}}({\mathcal {A}}^2)\). Since \({\mathfrak {X}}'\cap {\mathfrak {C}}=\emptyset \), the Hahn–Banach Theorem implies that there is a bounded linear functional \({\mathcal {L}}\) on \({\mathfrak {M}}({\mathcal {A}}^2)\) such that \({\mathcal {L}}(\mu )=0\) for all \(\mu \in {\mathfrak {C}}\), and \({\mathcal {L}}(\mu )<0\) for all \(\mu \) in the interior of \({\mathfrak {X}}'\), in particular for every \(\mu \in {\mathfrak {X}}\).
The first condition on \({\mathcal {L}}\) implies, by representation (12), that there is a potential function \(F(x,y)=g(x)-g(y)\) (with a bounded and measurable function \(g:~J\rightarrow {\mathbb {R}}\)) such that \({\mathcal {L}}(\mu )=\mu (F)\) for every \(\mu \in {\mathfrak {M}}({\mathcal {A}}^2)\) such that \(\mu \ll \psi \). Let \(|g|\le C\).
Let \(S=\{(x,y):~g(x)>g(y)\}\) and \(A_t=\{x\in J:~g(x)\ge t\}\). Clearly \(A_t\times A_t^c\subseteq S\) and \(A_t^c\times A_t\subseteq S^c\). We can write
$$\begin{aligned} g(x) = \int \limits _{-C}^C {\mathbb {1}}_{A_t}(x)\,dt, \end{aligned}$$
then
$$\begin{aligned} {\mathcal {L}}(\mu ) = \int \limits _{-C}^C \int \limits _{J\times J} {\mathbb {1}}_{A_t}(x)-{\mathbb {1}}_{A_t}(y)\,d\mu (x,y)\,dt = \int \limits _{-C}^C \mu (A_t\times A_t^c)-\mu (A_t^c\times A_t)\,dt. \end{aligned}$$
(14)
Let us apply this formula with \(\mu (X)=\varphi (X\cap S)+\psi (X\setminus S)\). Then
$$\begin{aligned} {\mathcal {L}}(\mu ) = \int \limits _{-C}^C \mu (A_t\times A_t^c)-\mu (A_t^c\times A_t)\,dt =\int \limits _{-C}^C \psi (A_t\times A_t^c)-\varphi (A_t^c\times A_t)\,dt \ge 0 \end{aligned}$$
by hypothesis. On the other hand, we have \(\varphi \le \mu \le \psi \), so \(\mu \in {\mathfrak {X}}\), so \({\mathcal {L}}(\mu )<0\). This contradiction proves (13).
To conclude, we select circulations \(\alpha _n\in {\mathfrak {C}}\) and measures \(\beta _n\in {\mathfrak {X}}\) such that \(\Vert \alpha _n-\beta _n\Vert \rightarrow 0\) (\(n\rightarrow \infty \)). By Lemma 3.1, there is a measure \(\beta \in {\mathfrak {X}}\) such that \(\beta _n(S)\rightarrow \beta (S)\) (\(n\rightarrow \infty \)) for all \(S\in {\mathcal {A}}^2\) and an appropriate subsequence of the indices n. Hence
$$\begin{aligned} |\alpha _n(S)-\beta (S)|\le & {} |\alpha _n(S)-\beta _n(S)|+|\beta _n(S)-\beta (S)|\\\le & {} \Vert \alpha _n-\beta _n\Vert +|\beta _n(S)-\beta (S)|\rightarrow 0. \end{aligned}$$
In particular, for every \(A\in {\mathcal {A}}\) we have
$$\begin{aligned} 0=\alpha _n(A\times A^c)-\alpha _n(A^c\times A)\rightarrow \beta (A\times A^c)-\beta (A^c\times A), \end{aligned}$$
and so \(\beta \) is a circulation, and by a similar argument, \(\beta \in {\mathfrak {X}}\). \(\square \)
Remark 4.5
As long as we restrict our attention to circulations \(\alpha \) that are absolutely continuous with respect to a given measure \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\), we can define them as functions, considering the Radon–Nikodym derivative \(f=d\alpha /d\psi \). Then f is a \(\psi \)-integrable function satisfying
$$\begin{aligned} \int \limits _{A\times A^c} f\,d\psi = \int \limits _{A^c\times A} f\,d\psi \end{aligned}$$
for all \(A\in {\mathcal {A}}\). The value f(x, y) can be interpreted as the flow value on the edge xy. The marginals of \(\alpha \), meaning the flow in and out of a point, could also be defined using a disintegration of \(\psi \). However, this definition of circulation would depend on the measure \(\psi \), while our definition above does not depend on any such parameter.
Similar remarks apply to notions like flows below, and will not be repeated.
Optimal circulations
If a feasible circulation exists, we may be interested in finding a feasible circulation \(\mu \) which minimizes a “cost”, or maximizes a “value” \(\mu (v)\), given by a bounded measurable function v on \(J\times J\). Equivalently, we want to characterize when a value of 1 (say) can be achieved. This cannot be characterized in terms of cut conditions any more, but an elegant necessary and sufficient condition can still be formulated.
Theorem 4.6
Given a bounded measurable function \(v:~J\times J\rightarrow {\mathbb {R}}_+\) and measures \(\varphi ,\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\), \(\varphi \le \psi \), there is a circulation \(\alpha \) with \(\varphi \le \alpha \le \psi \) and \(\alpha (v)=c\) if and only if the following three conditions are satisfied for every potential F:
$$\begin{aligned}&\psi (|F+v|_+) \ge \varphi (|F+v|_-) + c, \end{aligned}$$
(15)
$$\begin{aligned}&\psi (|F-v|_+) \ge \varphi (|F-v|_-) - c, \end{aligned}$$
(16)
$$\begin{aligned}&\psi (|F|_+) \ge \varphi (|F|_-). \end{aligned}$$
(17)
Condition (17) is equivalent to the condition given for the existence of a circulation in Theorem 4.4, which is obtained when \(F(x,y)={\mathbb {1}}_X(x)-{\mathbb {1}}_X(y)\). If \(\varphi =0\), then only (15) is nontrivial. Applying the conditions with \(F=0\) we get that \(\varphi (v)\le c\le \psi (v)\).
Proof
We may assume that \(c=1\). The necessity of the condition is trivial: if such a circulation \(\alpha \) exists, then
$$\begin{aligned} \psi (|F+v|_+) - \varphi (|F+v|_-) \ge \alpha (|F+v|_+) - \alpha (|F+v|_-)= \alpha (F+v) = \alpha (v) = 1, \end{aligned}$$
and similar calculation proves the other two conditions.
To prove the converse, we proceed along similar lines as in the proof of Theorem 4.4. Consider the subspace \({\mathfrak {C}}\subseteq {\mathfrak {M}}({\mathcal {A}}^2)\) of circulations, the affine hyperplane \({\mathfrak {H}}=\{\alpha \in {\mathfrak {M}}({\mathcal {A}}^2):~ \alpha (v)=1\}\) and the “box” \({\mathfrak {X}}=\{\alpha \in {\mathfrak {M}}({\mathcal {A}}^2):~ \varphi \le \alpha \le \psi \}\). We want to prove that \({\mathfrak {C}}\cap {\mathfrak {H}}\cap {\mathfrak {X}}\not =\emptyset \).
Clearly the sets \({\mathfrak {C}}\), \({\mathfrak {H}}\) and \({\mathfrak {X}}\) are nonempty. Fix an \(\varepsilon >0\), and replace them by their \(\varepsilon \)-neighborhoods \({\mathfrak {C}}'=\{\mu \in {\mathfrak {M}}({\mathcal {A}}^2):~d_\mathrm{tv}(\mu ,{\mathfrak {C}})<\varepsilon \}\) etc. We start with proving the weaker statement that
$$\begin{aligned} {\mathfrak {C}}'\cap {\mathfrak {H}}'\cap {\mathfrak {X}}'\not =\emptyset . \end{aligned}$$
(18)
Suppose not. Then Lemma 3.4 implies that there are bounded linear functionals \({\mathcal {L}}_1,{\mathcal {L}}_2,{\mathcal {L}}_3\) on \({\mathfrak {M}}({\mathcal {A}}^2)\), not all zero, and real numbers \(a_1,a_2,a_3\) such that \({\mathcal {L}}_1+{\mathcal {L}}_2+{\mathcal {L}}_3=0\), \(a_1+a_2+a_3=0\), and \({\mathcal {L}}_i(\mu )\ge a_i\) for all \(\mu \in {\mathfrak {C}}'\), \({\mathfrak {H}}'\) and \({\mathfrak {X}}'\), respectively, and \({\mathcal {L}}_i(\mu )>a_i\) for at least one i.
The functional \({\mathcal {L}}_1\) remains bounded from below for every circulation \(\alpha \in {\mathfrak {C}}\), and since \({\mathfrak {C}}\) is a linear subspace, this implies that
$$\begin{aligned} {\mathcal {L}}_1(\alpha )=0 \qquad (\alpha \in {\mathfrak {C}}). \end{aligned}$$
(19)
By a similar reasoning, \({\mathcal {L}}_2\) must be a constant b on the hyperplane \({\mathfrak {H}}\); we may scale \({\mathcal {L}}_1\), \({\mathcal {L}}2\) and \({\mathcal {L}}_3\) so that \(b\in \{-1,0,1\}\). It is easy to see that this implies the more general formula
$$\begin{aligned} {\mathcal {L}}_2(\mu ) = b\mu (v)\qquad (\mu \in {\mathfrak {M}}({\mathcal {A}}^2)), \end{aligned}$$
(20)
Finally, we can express \({\mathcal {L}}_3\) as
$$\begin{aligned} {\mathcal {L}}_3(\mu )=-{\mathcal {L}}_1(\mu )-{\mathcal {L}}_2(\mu )\quad (\mu \in {\mathfrak {M}}(A^2)). \end{aligned}$$
(21)
Using the representation (12), we can write
$$\begin{aligned} {\mathcal {L}}_1(\mu ) = \mu (F)\qquad (0\le \mu \le \psi ) \end{aligned}$$
(22)
with some potential F on \(J\times J\). Hence
$$\begin{aligned} {\mathcal {L}}_3(\mu )=-\mu (F)-b\mu (v)=-\mu (F+bv) \qquad (0\le \mu \le \psi ). \end{aligned}$$
We also know that for any \(\alpha \in {\mathfrak {C}}\), \(\nu \in {\mathfrak {H}}\) and \(\mu \in {\mathfrak {X}}\), we have
$$\begin{aligned} 0=a_1+a_2+a_3<{\mathcal {L}}_1(\alpha )+{\mathcal {L}}_2(\nu )+{\mathcal {L}}_3(\mu ) = 0+b+{\mathcal {L}}_3(\mu ) = b-\mu (F+bv), \end{aligned}$$
and hence \(\mu (F+bv)<b\) for all \(\mu \in {\mathfrak {X}}\).
The tightest choice for \(\mu \in {\mathfrak {X}}\) is \(\mu =\psi _U-\varphi _{U^c}\), where \(U=\{(x,y):~F(x,y)+bv(x,y)\ge 0\}\). This gives that
$$\begin{aligned} \psi (|F+bv|_+) - \varphi (|F+bv|_-) = \psi _U(F+bv) - \varphi _{U^c}(F+bv) = \mu (F+bv) <b. \end{aligned}$$
This contradicts one of the conditions in the theorem (depending on b). This proves (18).
To prove the stronger statement that \({\mathfrak {C}}\cap {\mathfrak {H}}\cap {\mathfrak {X}}\not =\emptyset \), (18) implies that there are sequences of measures \(\alpha _n\in {\mathfrak {C}}\), \(\nu _n\in {\mathfrak {H}}\) and \(\mu _n\in {\mathfrak {X}}\) such that \(d_\mathrm{tv}(\mu _n,\alpha _n)\rightarrow 0\) and \(d_\mathrm{tv}(\mu _n,\nu _n)\rightarrow 0\). Furthermore, since \(0\le \mu _n\le \psi \), Lemma 3.1 applies, and so there is a measure \(\mu \in {\mathfrak {X}}\) such that for an appropriate infinite subsequence of indices, \(\mu _n(U)\rightarrow \mu (U)\) for all \(U\in {\mathcal {A}}^2\). This implies that \(\alpha _n(U)\rightarrow \mu (U)\) and \(\nu _n(U)\rightarrow \mu (U)\) for this subsequence.
Thus
$$\begin{aligned} \mu (A\times A^c) = \lim _{n\rightarrow \infty } \alpha _n(A\times A^c) = \lim _{n\rightarrow \infty } \alpha _n(A^c\times A) = \mu (A^c\times A) \end{aligned}$$
for every \(A\in {\mathcal {A}}\), so \(\mu \in {\mathfrak {C}}\). Similarly, by Lemma 3.1\(\mu (v) = \lim _{n\rightarrow \infty }\nu _n(v) =1\), whence \(\mu \in {\mathfrak {H}}\). \(\square \)
A straightforward application of Theorem 4.6 allows us to answer a question about the existence of Markov spaces, where an upper bound on the ergodic circulation is prescribed.
Corollary 4.7
Given a measure \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\), there exists an ergodic circulation \(\eta \) such that \(\eta \le \psi \) if and only if every potential \(F:~J\times J\rightarrow {\mathbb {R}}\) satisfies
$$\begin{aligned} \psi (|1+F|_+) \ge 1. \end{aligned}$$
Integrality.
In the case when \(v\equiv 1\) and \(\varphi \equiv 0\), the condition in Corollary 4.7 implies that
$$\begin{aligned} \psi (A\times A^c)-\psi (A^c\times A)\le \psi (J\times J)-1 \qquad (A\in {\mathcal {A}}). \end{aligned}$$
One may wonder whether, at least in this special case, such a cut condition is also sufficient in Corollary 4.7. This, however, fails even in the finite case: on the directed path of length 2 where the edges have capacity 1, these cut conditions for the existence of an ergodic circulation are satisfied, but the only feasible circulation is the 0-circulation.
However, the following weaker requirement can be imposed on F:
Supplement 4.8
In Theorem 4.6, if the function v has only integral values, then it suffices to require condition (15)–(17) for potentials F having integral values.
This property of F is clearly equivalent to saying that in the representation \(F(x,y)=f(x)-f(y)\), the function f can be required to have integral values. For finite graphs, this assertion follows easily from the fact that the matrix of flow conditions is totally unimodular. In the infinite case, we have to use another proof.
Proof
Suppose that there is a potential \(F(x,y)=f(x)-f(y)\) violating (say) (15). Let \(S=\{(x,y):~F(x,y)+v(x,y)>0\}\). Consider the modified potentials \(\widehat{F}=\lfloor f(y)\rfloor -\lfloor f(y)\rfloor \) and \(\widetilde{F}=\langle f(x)\rangle -\langle f(y)\rangle \), where \(\langle t\rangle = t-\lfloor t\rfloor \) is the fractional part of the real number t. We claim that
$$\begin{aligned} \psi (|F+v|_+) - \varphi (|F+v|_-) = \psi (|\widehat{F}+v|_+) - \varphi (|\widehat{F}+v|_-) + \psi _S(\widetilde{F}) + \varphi _{S^c}(\widetilde{F}). \end{aligned}$$
(23)
Indeed, note that for \((x,y)\in S\) we have \(\widehat{F}(x,y)+v(x,y)\ge 0\), and for \((x,y)\notin S\) we have \(\widehat{F}(x,y)+v(x,y)\le 0\). Hence
$$\begin{aligned} \psi (|F+v|_+)= \psi _S(F+v) = \psi _S(\widehat{F}+v) + \psi _S(\widetilde{F})&= \psi (|\widehat{F}+v|_+) + \psi _S(\widetilde{F}). \end{aligned}$$
Similarly,
$$\begin{aligned} \varphi (|F+v|_-) = \varphi (|\widehat{F}+v|_-) - \varphi _{S^c}(\widetilde{F}). \end{aligned}$$
This proves (23).
Replacing f by \(f+a\) with any real constant a, the potential F and the set S do not change, but the potentials \(\widehat{F}_a(x,y)=\lfloor f(x)+a\rfloor -\lfloor f(y)+a\rfloor \) and \(\widetilde{F}_a(x,y) = \langle f(x)+a\rangle -\langle f(y)+a\rangle \) do depend on c. We have
$$\begin{aligned} \psi (|F+v|_+) - \varphi (|F+v|_-) = \psi (|\widehat{F}_a+v|_+)- \varphi (|\widehat{F}_a+v|_-) +\psi _S(\widetilde{F}_a)-\varphi _{S^c}(\widetilde{F}_a). \end{aligned}$$
Choosing a randomly and uniformly from [0, 1], the expectation of the last two terms is 0, since \(\mathsf{E}(\langle f(x)+a\rangle ) = 1/2\) for any x, and so \(\mathsf{E}(\widetilde{F}_a(x,y)) =0\) for all x and y. Thus
$$\begin{aligned} \psi (|F+v|_+) - \varphi (|F+v|_-) =\mathsf{E}\bigl (\psi (|\widehat{F}_a+v|_+) - \varphi (|\widehat{F}_a+v|_-)\bigr ). \end{aligned}$$
This implies that there is an \(a\in [0,1]\) for which
$$\begin{aligned} \psi (|F+v|_+) - \varphi (|F+v|_-) \ge \psi (|\widehat{F}_a+v|_+) - \varphi (|\widehat{F}_a+v|_-). \end{aligned}$$
So replacing f by \(\lfloor f+a\rfloor \), we get an integer valued potential that violates condition (15) even more, which proves the Supplement. \(\square \)
We can give a more combinatorial reformulation of Corollary 4.7.
Corollary 4.9
Given a measure \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\), there exists an ergodic circulation \(\eta \) such that \(\eta \le \psi \) if and only if for every partition \(J=S_1\cup \dots \cup S_k\) into a finite number of Borel sets
$$\begin{aligned} \sum _{1\le i\le j\le k} (j-i+1) \psi (S_j\times S_i) \ge 1. \end{aligned}$$
The (insufficient) cut condition discussed above corresponds to the case when \(k=2\).
Proof
Let \(F(x,y)=f(y)-f(x)\) be a bounded integral valued potential. We may assume that f is integral valued and \(1\le f\le k\) for some integer k. Then the sets \(S_i=\{x\in J:~f(x)=i\}\) \((i=1,\dots ,k)\) form a partition of J. For \(x\in S_i\) and \(y\in S_j\), we have
$$\begin{aligned} |F(x,y)+1|_+ = {\left\{ \begin{array}{ll} j-i+1, &{} \text {if }i\le j, \\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$
Thus the condition in Corollary 4.7 is equivalent to the condition in Corollary 4.9. \(\square \)
Flows.
Let \(\sigma ,\tau \in {\mathfrak {M}}({\mathcal {A}})\) be two measures with \(\sigma (J)=\tau (J)\). We consider \(\sigma \) the “supply” and \(\tau \), the “demand”. We call a measure \(\varphi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\) a flow from \(\sigma \) to \(\tau \), or briefly a \(\sigma \)-\(\tau \) flow, if \(\varphi ^1-\varphi ^2=\sigma -\tau \). We may assume, if convenient, that the supports of \(\sigma \) and \(\tau \) are disjoint, since subtracting \(\sigma \wedge \tau \) from both does not change their difference. If this is the case, we call \(\sigma (J)=\tau (J)\) the value of the flow.
Given two points \(s,t\in J\), a measure \(\varphi \) on \({\mathcal {A}}^2\) such that \(\varphi ^1-\varphi ^2 = a(\delta _s-\delta _t)\) will be called an s-t flow of value a. So \(\varphi \) is a flow serving supply \(a\delta _s\) and demand \(a\delta _t\).
Note that every measure \(\varphi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\) is a flow from \(\varphi ^1\) to \(\varphi ^2\), and also a flow from \(\varphi ^1\setminus \varphi ^2\) to \(\varphi ^2\setminus \varphi ^1\). But we are usually interested in starting with the supply and the demand, and constructing appropriate flows. We may require \(\varphi \) to be acyclic, since subtracting a circulation does not change \(\varphi ^1-\varphi ^2\).
As before, we may also be given a nonnegative measure \(\psi \) on \({\mathcal {A}}^2\) (the “edge capacity”). We call a flow \(\varphi \) feasible, if \(\varphi \le \psi \).
Max-Flow-Min-Cut and Supply-Demand.
These fundamental theorems follow from the results on circulations by the same tricks as in the finite case.
Theorem 4.10
(Max-Flow-Min-Cut). Given a capacity measure \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\) and two points \(s,t\in J\), there is a feasible s-t flow of value 1 if and only if \(\psi (A\times A^c)\ge 1\) for every \(A\in {\mathcal {A}}\) with \(s\in A\) and \(t\notin A\).
Proof
For every feasible flow \(\phi \le \psi \) of value 1, the measure \(\phi +\delta _{ts}\) is a circulation such that \(\delta _{st}\le \phi +\delta _{st}\le \psi +\delta _{st}\). Conversely, for every circulation \(\alpha \) with \(\delta _{ts}\le \alpha \le \psi +\delta _{st}\), the measure \(\alpha -\delta _{ts}\) is a feasible s-t flow of value 1. The conditions in Theorem 4.4 on the existence of such a circulation are trivial except for the second condition when \(s\in A\) and \(t\notin A\), which gives the condition in the theorem. \(\square \)
The more general Supply-Demand Theorem can be stated as follows.
Theorem 4.11
Let \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\), and let \(\sigma ,\tau \in {\mathfrak {M}}_+({\mathcal {A}})\) with \(\sigma (J)=\tau (J)\). Then there is a feasible \(\sigma \)-\(\tau \) flow if and only if \(\psi (S\times S^c)\ge \sigma (S)-\tau (S)\) for every \(S\in {\mathcal {A}}\).
Proof
We may assume that \(\sigma (J)=\tau (J)=1\). Add two new points s and t to J, and extend \({\mathcal {A}}\) to a sigma-algebra \({\mathcal {A}}'\) on \(J'=J\cup \{s,t\}\) generated by \({\mathcal {A}}\), \(\{s\}\) and \(\{t\}\). Define a new capacity measure \(\psi '\) by
$$\begin{aligned} \psi '(X)= {\left\{ \begin{array}{ll} \psi (X), &{} \text {if }X\subseteq J\times J, \\ \sigma (Y), &{} \text {if }X= \{s\}\times Y\text { with }Y\subseteq J, \\ \tau (Y), &{} \text {if }X= Y\times \{t\}\text { with }Y\subseteq J,\\ 0, &{} \text {if }X\subseteq (\{t\}\times J) \cup (J\times \{s\}) \cup \{st,ts\}, \end{array}\right. } \end{aligned}$$
and extend it to all Borel sets by additivity. For every feasible \(\sigma \)-\(\tau \) flow \(\phi \) on \((J,{\mathcal {A}})\), the measure \(\phi +\psi '_{\{s\}\times J}+\psi '_{J\times \{t\}}\) is a feasible s-t flow of value 1. Conversely, for every feasible s-t flow of value 1, its restriction to the original space \((J,{\mathcal {A}})\) is a feasible \(\sigma \)-\(\tau \) flow. Applying the condition in the Max-Flow-Min-Cut Theorem completes the proof. \(\square \)
The measure-theoretic Max-Flow-Min-Cut Theorem is closely related to a result of Laczkovich [18], who works in the function setting. He also states an integrality result, which is in a sense dual to our integrality result in Section 4.2.4.
A condition for the minimum cost of a feasible \(\sigma \)-\(\tau \) flow of a given value can be derived from Theorem 4.6 using the same kind of constructions as in the proof above. This gives the following result.
Theorem 4.12
Given a bounded measurable “cost” function \(v:~J\times J\rightarrow {\mathbb {R}}_+\), a “capacity” measure \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\) and “supply-demand” measures \(\sigma ,\tau \in {\mathfrak {M}}_+({\mathcal {A}})\) with \(\sigma (J)=\tau (J)\), there is a feasible \(\sigma \)-\(\tau \) flow \(\varphi \) with \(\varphi (v)=1\) if and only if
$$\begin{aligned} \psi (|f(y)-f(x)+bv(x,y)|_+) \ge \tau (f)-\sigma (f)+b \end{aligned}$$
(24)
for every bounded measurable function \(f:~J\rightarrow {\mathbb {R}}\) and \(b\in \{-1,0,1\}\). \(\square \)
Transshipment.
An optimization problem closely related to flows is the transshipment problem. In its simplest measure-theoretic version, we are given two measures \(\alpha ,\beta \in {\mathfrak {M}}({\mathcal {A}})\) with \(\alpha (J)=\beta (J)\). An \(\alpha \)-\(\beta \) transshipment is a measure \(\mu \in {\mathfrak {M}}_+({\mathcal {A}}\times {\mathcal {A}})\) coupling \(\alpha \) and \(\beta \); in other words, \(\mu ^1=\alpha \) and \(\mu ^2=\beta \). Note the difference with the notion of an \(\alpha \)-\(\beta \) flow: there only the difference \(\mu ^1-\mu ^2\) is prescribed. In transhipment problems, one can think of \(J\times J\) as the edge set of a (complete) bipartite graph whose color classes are the two copies of J. This observation can be used to derive the following result from the Supply-Demand Theorem 4.11:
Theorem 4.13
Let \((J,{\mathcal {A}})\) be a standard Borel space, and \(\alpha ,\beta \in {\mathfrak {M}}_+({\mathcal {A}})\) with \(\alpha (J)=\beta (J)\). Let \(\psi \in {\mathfrak {M}}_+({\mathcal {A}}\times {\mathcal {A}})\). Then there exists an \(\alpha \)-\(\beta \) transshipment \(\mu \) with \(\mu \le \psi \) if and only if
$$\begin{aligned} \psi (S\times T)\ge \alpha (S)+\beta (T)-\alpha (J) \end{aligned}$$
for every \(S,T\in {\mathcal {A}}\). \(\square \)
Suppose that every edge \((x,y)\in J\times J\) has a given cost \(c(x,y)\ge 0\). We want to find a transshipment minimizing the cost \(\mu (c)\). We note that the minimum is attained by Lemma 3.2.
Theorem 4.14
Let \((J,{\mathcal {A}})\) be a standard Borel space, and \(\alpha ,\beta \in {\mathfrak {M}}_+({\mathcal {A}})\) with \(\alpha (J)=\beta (J)\). Let \(c:~J\times J\rightarrow {\mathbb {R}}_+\) be a bounded measurable function. Then the minimum cost of an \(\alpha \)-\(\beta \) transshipment is \(\sup _{g,h} \alpha (g)+\beta (h)\), where g and h range over all bounded measurable functions \(J\rightarrow {\mathbb {R}}\) satisfying \(g(x)+h(y)\le c(x,y)\) for all \(x,y\in J\).
The proof follows by an easy reduction to Theorem 4.12.
As a third variation on the Transshipment Problem, we ask for a transhipment supported on a specified set E of pairs. The following result is a slight generalization of a theorem of Strassen [26], and essentially equivalent to Proposition 3.8 of Kellerer [16]. See also [9]. It is also a rather straightforward generalization of Theorem 2.5.2 in [22]. The result could also be considered as a limiting case of Theorem 4.14, using the capacity “measure” with infinite values on E.
Proposition 4.15
Let \((J,{\mathcal {A}})\) be a standard Borel space, and \(\alpha ,\beta \in {\mathfrak {M}}_+({\mathcal {A}})\) with \(\alpha (J)=\beta (J)=1\). Let \(E\in {\mathcal {A}}\times {\mathcal {A}}\) be a Borel set such that \(J\times J\setminus E\) is the union of a countable number of product sets \(A\times B\) \((A,B\in {\mathcal {A}})\). Then there exists an \(\alpha \)-\(\beta \) transshipment \(\mu \) concentrated on E if and only if \(\alpha (S)+\beta (T)\le 1\) for any two sets \(S,T\in {\mathcal {A}}\) with \(S\times T\cap E=\emptyset \).
Remark 4.16
In the finite case, the fundamental Birkhoff–von Neumann Theorem describes the extreme points of the convex polytope formed by doubly stochastic matrices: these are exactly the permutation matrices, or in the language of bipartite graphs, perfect matchings. One generalization of this problem to the measurable case is to consider the set of coupling measures between two copies of a probability space \((J,{\mathcal {A}},\pi )\), forming a convex set in \({\mathfrak {M}}_+({\mathcal {A}}^2)\). What are the extreme points (coupling measures) of this convex set? Unfortunately, these extreme points seem to be too complex for an explicit description. See [19] for several examples.
Path decomposition.
In finite graph theory, it is often useful to decompose an s-t flow into a convex combination of flows along single paths from s to t and circulations along cycles. We will also need a generalization of this construction to measurable spaces.
Let \(K=J\cup J^2\cup J^3\cup \dots \) be the set of all finite nonempty sequences of points of J; we also call these walks. The set K is endowed with the sigma-algebra \({\mathcal {B}}={\mathcal {A}}\oplus {\mathcal {A}}^2 \oplus \dots \). Let K(s, t) be the subset of K consisting of walks starting at s and ending at t (\(s,t\in J\)); such a walk is called an s-twalk.
Let \(\tau \in {\mathfrak {M}}_+({\mathcal {B}})\). For \(Q=(u^0,u^1,\dots ,u^m)\in K\), let \(Q'=(u^0,\dots ,u^{m-1})\), \(V(Q)=\{u^0,\dots ,u^m\}\), \(E(Q)=\{u^0u^1,u^1u^2,\dots ,u^{m-1}u^m\}\), and \(Z(Q)=\{u^0,u^m\}\). Define
$$\begin{aligned} V(\tau )(X)&= \int \limits _K |V(Q')\cap X|\,d\tau (Q) \qquad (X\in {\mathcal {A}}),\\ E(\tau )(Y)&= \int \limits _K |E(Q)\cap Y|\,d\tau (Q) \qquad (Y\in {\mathcal {A}}^2),\\ Z(\tau )(Y)&= \int \limits _K |Z(Q)\cap Y|\,d\tau (Q) \qquad (Y\in {\mathcal {A}}^2). \end{aligned}$$
Then \(V(\tau )\) is a measure on \({\mathcal {A}}\), and \(E(\tau )\) and \(Z(\tau )\) are measures on \({\mathcal {A}}^2\). The measure \(Z(\tau )\) is finite, but \(V(\tau )\) and \(E(\tau )\) may have infinite values as for now. If \(\tau \) is a probability measure, then walking along a randomly chosen walk from distribution \(\tau \), \(V(\tau )(X)\) is the expected number of times we exit a point in X (so the starting point counts, but the last point does not), and \(E(\tau )(Y)\) is the expected number of times we traverse an edge in Y. Mapping each walk \(W\in K\) to its first point, and pushing \(\tau \) forward by this map, we get the measure \(Z(\tau )^1\in {\mathfrak {M}}({\mathcal {A}})\). The measure \(Z(\tau )^2\) is characterized analogously by mapping each walk to its last point. It is easy to see that \(E(\tau )\) is a flow from \(Z(\tau )^1\) to \(Z(\tau )^2\).
Theorem 4.17
For every acyclic measure \(\varphi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\) there is a finite measure \(\tau \in {\mathfrak {M}}_+({\mathcal {B}})\) for which \(E(\tau )=\varphi \).
We need a simple (folklore) fact about Markov chains.
Lemma 4.18
Let \({\mathbf {G}}\) be an indecomposable Markov space, and let \(S\in {\mathcal {A}}\) have \(\pi (S)>0\). Then for \(\pi \)-almost-all starting points x, a random walk started at x hits S almost surely.
Proof
Let R be the set of starting points \(x\in J\) for which the random walk starting at x avoids S with positive probability, and suppose that \(\pi (R)>0\). Since clearly \(R\cap S=\emptyset \), we also have \(\pi (R)<1\). Hence \(\eta (R^c\times R)>0\) by indecomposability, and so there must be a point \(x\in R^c\) with \(P_x(R)>0\). But this means that starting at x, the walk moves to R with positive probability, and then avoids S with positive probability, so we would have \(x\in R\), a contradiction. \(\square \)
Proof of Theorem 4.17
We start with the special case when \(\varphi \) is an s-t flow for \(s,t\in J\); we may scale it to have value 1. Just as in the proof of Theorem 4.10, we see that the measure \(\alpha =\varphi + \delta _{ts}\) is a nonnegative circulation on \({\mathcal {A}}^2\). Let \(a=\alpha (J\times J) = \varphi (J\times J)+1\), then \(\eta =\alpha /a\) is the ergodic circulation of a Markov space. The stationary distribution of this Markov space is \(\pi =\alpha ^1/a=\alpha ^2/a\), and
$$\begin{aligned} \varphi ^1 = a\pi - \delta _t. \end{aligned}$$
(25)
It is easy to see that \(\varphi (\{(s,s)\})=0\), since \(\xi =\varphi (\{(s,s)\})\delta _{\{(s,s)\}}\) is a nonnegative circulation such that \(\xi \le \varphi \), and since \(\varphi \) is acyclic, we must have \(\xi =0\).
Claim 1
The Markov space \(({\mathcal {A}},\eta )\) is indecomposable.
Indeed, suppose that there is a set \(A\in {\mathcal {A}}\) with \(0<\pi (A)<1\) and \(\eta (A\times A^c)=\eta (A^c\times A)=0\). Clearly s and t either both belong to A or both belong to \(A^c\); we may assume that \(s,t\in A^c\). Then \(\varphi _{A\times A}\) is a circulation, and \(\varphi =(\varphi -\varphi _{A\times A})+\varphi _{A\times A}\) is a decomposition showing that \(\varphi \) is not acyclic, contrary to the hypothesis.
To specify a probability distribution on s-t walks, we describe how to generate a random s-t walk: Start a random walk at s, and follow it until you hit t or return to s, whichever comes first. This happens almost surely by Lemma 4.18: the distribution \(\delta _s\) is absolutely continuous with respect to \(\pi \), and \(\pi (t)>0\). This gives a probability distribution \(\tau \) on the set \(K(s,\{s,t\})\) of walks from s to \(\{s,t\}\).
Let us stop the walk after k steps, or when it hits t, or when it returns to s, whichever comes first. This gives us a distribution \(\tau _k\) over walks starting at s of length at most k. We claim that this distribution satisfies the following identity for every \(X\subseteq J\setminus \{s,t\}\):
$$\begin{aligned} V(\tau _n)(X)= \int \limits _{J\setminus \{s,t\}} P_u(X)\,dV(\tau _{n-1})(u). \end{aligned}$$
(26)
Indeed, let \(\sigma _k(X)\) \((X\in {\mathcal {A}})\) be the probability that starting at s, we walk k steps without hitting t or returning to s, and after k steps we are in X. It is clear that \(\sigma _0=\delta _s\). It is also easy to see that for \(n\ge 1\), we have \(V(\tau _n)=\sigma _0+\sigma _1+\dots +\sigma _{n-1}\), and for \(X\subseteq J\setminus \{s,t\}\),
$$\begin{aligned} \sigma _n(X)= \int \limits _{J\setminus \{t\}} P_u(X)\,d\sigma _{n-1}(u). \end{aligned}$$
(27)
Thus
$$\begin{aligned} V(\tau _n)(X) = \sum _{k=1}^{n-1} \sigma _k(X) = \sum _{k=1}^{n-1} \int \limits _{J\setminus \{t\}} P_u(X)\,d\sigma _{k-1}(u) = \int \limits _{J\setminus \{t\}} P_u(X)\,dV(\tau _{n-1})(u). \end{aligned}$$
This proves (26).
Next we show that
$$\begin{aligned} V(\tau _n) \le \varphi ^1\qquad (n\ge 1). \end{aligned}$$
(28)
We prove the inequality by induction on n. For \(n=1\) it is obvious. Let \(n\ge 2\). If \(s,t\notin X\), then \(\sigma _0(X)=0\), and so using (26) and (25),
$$\begin{aligned} V(\tau _n)(X)&= \int \limits _{J\setminus \{t\}} P_u(X)\,dV(\tau _{n-1})(u)\\&\le \int \limits _{J\setminus \{t\}} P_u(X)\,d\varphi ^1(u)\le a\int \limits _{J\setminus \{t\}} P_u(X)\,d\pi (u) \\&\le a\int \limits _J P_u(X)\,d\pi (u) = a\pi (X)= \varphi ^1(X). \end{aligned}$$
If \(t\in X\) but \(s\notin X\), then
$$\begin{aligned} V(\tau _n)(X) = V(\tau _n)(X\setminus \{t\}) \le \varphi ^1(X\setminus \{t\}) \le \varphi ^1(X). \end{aligned}$$
If \(s\in X\), then (using that every random walk we constructed exits s only once)
$$\begin{aligned} V(\tau _n)(X) = 1+V(\tau _n)(X\setminus \{s\}) \le 1+\varphi ^1(X\setminus \{s\})\le \varphi ^1(X). \end{aligned}$$
Next, we consider \(E(\tau )\), which is an s-t flow by the discussion before the theorem. It follows easily that
$$\begin{aligned} E(\tau _n) \le \varphi \qquad (n\ge 1). \end{aligned}$$
(29)
Indeed, for \(A,B\in {\mathcal {A}}\),
$$\begin{aligned} E(\tau _n)(A\times B) = \int \limits _A P_u(B)\,dV(\tau _n)(u) \le \int \limits _A P_u(B)\,d\varphi ^2(u)=\varphi (A\times B). \end{aligned}$$
This implies that \(E(\tau _n)(X)\le \varphi (X)\) for every \(X\in {\mathcal {A}}^2\), proving (29).
Claim 2
\(V(\tau _n)\rightarrow V(\tau )\) in total variation distance.
Since clearly \(V(\tau _n)\le V(\tau )\), we have \(d_\mathrm{tv}(V(\tau _n),V(\tau ))=V(\tau )(J)-V(\tau _n)(J)\). Let \(p_n\) be the probability that a random walk started at s first hits \(\{s,t\}\) in exactly n steps. Then
$$\begin{aligned} V(\tau )(J)=\sum _{k=1}^\infty p_k\,k, \qquad \text {and}\qquad V(\tau _n)(J)=\sum _{k=1}^n p_k\,k. \end{aligned}$$
By (28), \(V(\tau _n)(J)\le \varphi ^1(J)<\infty \), and hence the series representing \(\tau \) is convergent. This proves the claim.
Claim 3
The probability that a random walk started at s returns to s before hitting t is zero. So \(\tau \) can be considered as a probability distribution on walks from s to t.
Indeed, we can split \(K(s,\{s,t\})=K(s,s)\cup K(s,t)\). Define \(\rho =\tau _{K(s,s)}\). Then \(E(\rho )\le E(\tau )\le \varphi \) and it is easy to see that \(E(\rho )\) is a circulation. Since \(\varphi \) is acyclic, we must have \(\rho =0\), and so \(\tau (K(s,s))=0\).
Inequalities (28), (29) and Claim 2 imply that \(V(\tau )\le \varphi ^1\) and \(E(\tau ) \le \varphi \). To complete the proof, consider the measure \(\varphi -E(\tau )\). This is a nonnegative circulation, and since \(\varphi \) is acyclic, it follows that \(\varphi -E(\tau )=0\). This proves the theorem for s-t flows.
The general case can be reduced to the special case of an s-t flow by the following construction, similar to that used in the proof of Theorem 4.11. Let \(\varphi \in {\mathfrak {M}}_+({\mathcal {A}}^2)\) be an acyclic measure, let \(\sigma =\varphi ^1\setminus \varphi ^2\) and \(\tau =\varphi ^2\setminus \varphi ^1\), so that \(\varphi \) is an acyclic \(\sigma \)-\(\tau \) flow. Create two now points s and t, extend \({\mathcal {A}}\) to a sigma-algebra \({\mathcal {A}}'\) on \(J'=J\cup \{s,t\}\) generated by \({\mathcal {A}}\), \(\{s\}\) and \(\{t\}\), and extend the measure \(\varphi \) to \(\varphi '\in {\mathfrak {M}}({\mathcal {A}}'\times {\mathcal {A}}')\) by
$$\begin{aligned} \varphi '(X)= {\left\{ \begin{array}{ll} \varphi (X), &{} \text {if }X\subseteq J\times J, \\ \sigma (Y), &{} \text {if }X= \{s\}\times Y\text { with }Y\subseteq J, \\ \tau (Y), &{} \text {if }X= Y\times \{t\}\text { with }Y\subseteq J,\\ 0, &{} \text {if }X\subseteq (\{t\}\times J) \cup (J\times \{s\}) \cup \{st,ts\}. \end{array}\right. } \end{aligned}$$
It is easy to check that \(\varphi '\) is an acyclic s-t flow. Using the theorem for the special case of this s-t flow, we get a measure \(\tau \) on s-t paths, in which the trivial path (s, t) has zero measure. So \(\tau \) defines a measure on nontrivial s-t paths, and since there is a natural bijection with paths in K, we get a measure on \((K,{\mathcal {B}})\). It is easy to check that this measure has the desired properties. \(\square \)
Remark 4.19
Theorem 4.17 raises the question whether circulations have analogous decompositions. In finite graph theory, a circulation can be decomposed into a nonnegative linear combination of directed cycles. In the infinite case, we have to consider, in addition, directed paths infinite in both directions (see Example 4.1); but even so, the decomposition is not well understood.
Suppose that we have a nonnegative circulation \(\eta \not =0\) on \({\mathcal {A}}\). We may assume (by scaling) that it is a probability measure, so it is the ergodic circulation of a Markov space. From every point \(u\in J\), we can start an infinite random walk \((v^0=u, v^1,\dots )\), and also an infinite random walk \((v^0=u, v^{-1},\dots )\) of the reverse chain. Choosing u from \(\pi \), this gives us a probability distribution \(\beta \) on rooted two-way infinite (possibly periodic) sequences, i.e., on \(J^{{\mathbb {Z}}}\). However, it seems to be difficult to reconstruct the circulation \(\alpha \) from \(\beta \).