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### A categorical approach to integration.

Theory and Applications of Categories [electronic only]

### A Choquet Ordering and Unique Decompositions in Convex sets of Tight Measures

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

### A probabilistic ergodic decomposition result

Annales de l'I.H.P. Probabilités et statistiques

Let $\left(X,𝔛,\mu \right)$ be a standard probability space. We say that a sub-σ-algebra $𝔅$ of $𝔛$decomposes μ in an ergodic way if any regular conditional probability ${}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P$ with respect to $𝔅$ andμ satisfies, for μ-almost every x∈X, $\forall B\in 𝔅,{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,B\right)\in \left\{0,1\right\}$. In this case the equality $\mu \left(·\right)={\int }_{X}{}^{𝔅}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\left(x,·\right)\mu \left(\mathrm{d}x\right)$, gives us an integral decomposition in “$𝔅$-ergodic” components. For any sub-σ-algebra $𝔅$ of $𝔛$, we denote by $\overline{𝔅}$ the smallest sub-σ-algebra of $𝔛$ containing $𝔅$ and the collection of all setsAin $𝔛$ satisfyingμ(A)=0. We say that $𝔅$ isμ-complete if $𝔅=\overline{𝔅}$. Let $\left\{{𝔅}_{i}i\in I\right\}$ be a non-empty family...

### A uniqueness result for the continuity equation in two dimensions

Journal of the European Mathematical Society

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\stackrel{˙}{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with bounded divergence....

### An exact functional Radon-Nikodym theorem for Daniell integrals

Studia Mathematica

Given two positive Daniell integrals I and J, with J absolutely continuous with respect to I, we find sufficient conditions in order to obtain an exact Radon-Nikodym derivative f of J with respect to I. The procedure of obtaining f is constructive.

### Application de l'existence d'un relèvement à un théorème sur la désintégration des mesures

Annales de l'I.H.P. Probabilités et statistiques

### Characterization of optimal shapes and masses through Monge-Kantorovich equation

Journal of the European Mathematical Society

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

### Correction to the paper "Two classes of measures''

Colloquium Mathematicae

### Décomposition des mesures selon la dimension

Colloquium Mathematicae

### Décomposition et classification des systèmes dynamiques

Bulletin de la Société Mathématique de France

### Desintegration and compact measures.

Mathematica Scandinavica

### Désintégration d'une mesure

Séminaire Équations aux dérivées partielles (Polytechnique)

### Désintégration d'une mesure non bornée

Annales de l'I.H.P. Probabilités et statistiques

### Désintégration régulière d'une mesure par rapport à une famille de tribus

Séminaire Équations aux dérivées partielles (Polytechnique)

### Disintegration and perfectness of measure spaces.

Manuscripta mathematica

### Disintegration theory on a constant field of non-seperable Hilbert spaces.

Mathematica Scandinavica

### Flows in infinite networks

Proceedings of the 11th Winter School on Abstract Analysis

### Induced ...-Homomorphisms and a Parametrization of Measurable Sections via Extremal Preimage Measures.

Mathematische Annalen

### Integration seperat stetiger Funktionen

Mathematica Scandinavica

### Lifting and Desintegration

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

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