The paper by Jürgen Moser discusses the properties and equivalence of volume elements on a closed, connected $n$-dimensional manifold $M$. A volume element is defined as an $n$-form of odd kind that is everywhere positive. Two volume elements $\tau$ and $\sigma$ are considered equivalent if they can be transformed into each other by a $C^\infty$- automorphism of $M$. The main theorem states that if $\tau$ and $\sigma$ are two volume elements on a compact, connected manifold, then their total volumes are invariant under such automorphisms, and this volume is the only invariant. Moser proves this theorem by reducing it to a local statement where the function $f$ that differs from 1 in one coordinate patch only. He uses a partition of unity and a specific construction to decompose the function $f - 1$ into functions with support in disjoint patches. This allows him to construct a diffeomorphism that maps the volume element $\sigma$ to $\lambda \tau$, where $\lambda$ is a constant determined by the integrals of the two volume elements. The proof involves two lemmas. Lemma 1 shows that any continuous function $g$ on $M$ with $\int_M g \tau = 0$ can be decomposed into functions $g_j$ with support in disjoint patches and satisfying $\int g_j \tau = 0$. Lemma 2 provides a coordinate transformation that maps the unit cube to itself and satisfies the required conditions for the volume elements. Moser also presents an alternative proof using Hodge's decomposition theorem for forms on Riemannian manifolds. This proof extends the result to closed 2-forms and shows that if a family of closed 2-forms or $n$-forms of odd kind are nondegenerate and have fixed periods, then there exists an automorphism that transforms them into a common form. The paper concludes with a discussion on the difficulty of deciding when two closed, nondegenerate 2-forms belong to the same cohomology class and can be deformed into each other within that class.The paper by Jürgen Moser discusses the properties and equivalence of volume elements on a closed, connected $n$-dimensional manifold $M$. A volume element is defined as an $n$-form of odd kind that is everywhere positive. Two volume elements $\tau$ and $\sigma$ are considered equivalent if they can be transformed into each other by a $C^\infty$- automorphism of $M$. The main theorem states that if $\tau$ and $\sigma$ are two volume elements on a compact, connected manifold, then their total volumes are invariant under such automorphisms, and this volume is the only invariant. Moser proves this theorem by reducing it to a local statement where the function $f$ that differs from 1 in one coordinate patch only. He uses a partition of unity and a specific construction to decompose the function $f - 1$ into functions with support in disjoint patches. This allows him to construct a diffeomorphism that maps the volume element $\sigma$ to $\lambda \tau$, where $\lambda$ is a constant determined by the integrals of the two volume elements. The proof involves two lemmas. Lemma 1 shows that any continuous function $g$ on $M$ with $\int_M g \tau = 0$ can be decomposed into functions $g_j$ with support in disjoint patches and satisfying $\int g_j \tau = 0$. Lemma 2 provides a coordinate transformation that maps the unit cube to itself and satisfies the required conditions for the volume elements. Moser also presents an alternative proof using Hodge's decomposition theorem for forms on Riemannian manifolds. This proof extends the result to closed 2-forms and shows that if a family of closed 2-forms or $n$-forms of odd kind are nondegenerate and have fixed periods, then there exists an automorphism that transforms them into a common form. The paper concludes with a discussion on the difficulty of deciding when two closed, nondegenerate 2-forms belong to the same cohomology class and can be deformed into each other within that class.