Let $A$ be a non atomic measure on $\mathbb R$. Consider the product measure $\mu := A \times \dots \times A$ on $\mathbb R^n$.


**Question:** Let $M$ be a $n-1$ dimensional smooth submanifold of $\mathbb R^n$. Is it true that $M$ has measure $0$ under $\mu$?



Does a submanifold of nonzero codimension have measure zero under the product of non atomic measures?
A submanifold $M$ of ${\mathbb R}^{n}$ (with codimension 1) can always (after permuting the coordinates) be locally represented in the form $G\_f=\{(x,y) : y=f(x)\}$ where $x$ runs over $V \subset {\mathbb R}^{n-1}$ and $f:V \to {\mathbb R}$. That implies that $M$ is contained in a countable union of such graphs. Each graph has measure zero under the given product measure by Fubini since $A$ is nonatomic. Thus $M$ indeed has measure zero under the product measure by countable additivity.


A submanifold $M$ of ${\mathbb R}^{n}$ with positive codimension is always a subset of a manifold of codimension 1.


