This is not an answer, but on reading the discussion I thought that it would be nice if someone gave the definition of a density so that no one would think that it is a complicated object. I learned the following (somewhat more general definition) from I.M. Gelfand:
Definition.
A $k$-density on a manifold $M$ is a continuous real-valued function defined on the
cone of simple (a.k.a. decomposable) tangent $k$-vectors on $M$ that is homogeneous of degree one. A $k$-density $\varphi$ is said to be smooth if for every $k$-tuple of smooth linearly independent vector fields $X_i$ $(1 \leq i \leq k)$ defined in some open set $U \subset M$, we have that the function
$$
y \mapsto \varphi(X_1(y)\wedge \cdots \wedge X_k(y))
$$
is smooth in $U$.
A densitiy is called even if $\varphi(-v) = \varphi(v)$ for every simple tangent $k$-vector $v$. Likewise, we have odd $k$-densities that generalize differential $k$-forms
Examples and context
If $\Omega$ is a volume form on a manifold of dimension $n$, then both $\Omega$ and $|\Omega|$ are $n$-densities. The arc-length element of a Riemannian or Finsler metric is a $1$-density. The $k$-area integrand of a Riemannian or Finsler manifold is a $k$-density.
Parametric integrands (in the sense of Federer-Fleming) define $k$-densities when restricted to the cone of simple vectors, but densities are way more general.
Varifolds of dimension $k$ are elements of the dual to the space of even $k$-densities. This is basically their definition: because of their homogeneity even $k$-densities can be seen as continuous functions on the bundle of tangent $k$-planes.
A message from our sponsor
For more examples and for neat applications to integral geometry (if I may say so myself ...), which becomes much easier if differential forms are replaced by densities, see the paper Gelfand transforms and Crofton formulas.
