Let $(X,𝔛,\mu )$ 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 ${}^{𝔅}P$ with respect to $𝔅$ andμ satisfies, for μ-almost every x∈X, $\forall B\in 𝔅,{}^{𝔅}P(x,B)\in \{0,1\}$. In this case the equality $\mu (·)={\int }_X{}^{𝔅}P(x,·)\mu \left(\mathrm{d}x\right)$, 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 $\{{𝔅}_{i}i\in I\}$ be a non-empty family...

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ 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....

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.