Volume preserving embeddings of open subsets of ${\mathbb R}^n$ into manifolds

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $\Omega$, and we show that an open subset $U$ of ${\mathbb R}^n$of Lebesgue measure $\operatorname{Vol}\; (U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\operatorname{Vol}\; (U) \le \operatorname{Vol}\;(M, \Omega)$ is met.