arbitrary distribution may $\eta$ may arise as the derivative of a family of measures, $d\mu_t/dt|_{t=t_0}=\eta$, and we can try to use the continuity equation to try to assign an object that will give an idea of direction of the movement determined by $\eta$.

The assignment of an object conveying the direction of movement and analogous to $v_t$ can be done using Colombeau algebras. These algebras were developed [@colombeau] to provide a context in which distributions could be multiplied. The spaces of distributions are subsets of these algebras. Roughly speaking, the solution to the multiplication problem is to record, instead of the distribution itself, all possible smoothings of the distribution, because there is no difficulty in multiplying smooth densities. As we will explain below, an equivalence relation is then introduced on a certain set of families of smooth functions, and its equivalence classes are the elements of the algebra.

#### Construction.

To define the relevant Colombeau algebra, we follow [@colombeaubook Section 8.5]. Let $\mathcal E({\mathbb{R}}^n)$ be the set of families $(f_\varepsilon)_{0<\varepsilon<1}$ of functions $f_\varepsilon\in C^\infty({\mathbb{R}}^n)$ indexed by $0<\varepsilon<1$, such that for each compact set $K\subset {\mathbb{R}}^n$ and every multi-index $I$ there are $N\in {\mathbb{N}}$, $\gamma>0$, and $c>0$ such that $$\sup_{x\in K}\left|\partial^If_\varepsilon(x)\right|{\leqslant}\frac{c}{\varepsilon^N}\quad\textrm{if 