# Zariski closed sets in C^n are of measure 0

This is related to another question in which it is proved that Zariski open sets are dense in analytic topology.

But it is intuitive that something more is true. Namely, that they are the sets where some polynomials vanish, and consideration of a few examples in $\mathbb R^n$ where they are of Lebesgue measure $0$, suggest strongly that the Zariski-closed sets(except the whole affine space) are of measure $0$ in $\mathbb C^n$ as well. This should be quite simple; but I am unable to prove it due to inexperience in measure theory.

The nice thing about proving this is that once this is done, then we are able to claim safely that so-and-so statement is true almost everywhere, if it is true on a Zariski-open set.

So, in a more measure theoretic formulation:

Let $X$ be a set in $\mathbb C^n$ contained in the zero locus of some collection of polynomials. How to show that $X$ is of measure $0$?

In fact my feeling is that more should be true, ie, we can replace polynomials by analytic functions at least, and get the same result.

• Three out of four ain't bad! (Hint: an s is missing in the title.) I've batted worse with someone else's name in a paper. May 21, 2010 at 20:45
• Thanks Willie Wong! I am sorry to have messed it up like this. May 21, 2010 at 21:33
• Induct on $n$. When $n = 1$, this is true since zeros of a non trivial polynomial is a finite set. For inductive step use Fubini's theorem. Nov 27, 2010 at 0:19

If a real analytic function $f:U\subset\mathbb R^n\to\mathbb R^m$ is zero on a set $Z$ of positive measure (and $U$ is connected), then $f\equiv 0$.

Indeed, almost every point of $Z$ is a density point. It is easy to see that the derivative at a density point is zero. Therefore $df=0$ a.e. on $Z$. Applying the same argument to $df$, conclude that the second derivative vanishes a.e. on $Z$ too. And so on. Thus $f$ has zero Taylor expansion at some point, hence $f\equiv 0$.

• is there any officially published paper or book that state this property? I need it for referencing. Dec 20, 2019 at 18:15
• The link in the answer is broken. I would also like a reference to cite for such questions. Jul 20, 2020 at 16:14
• The link was just double escaped. It's Lebesgue's density theorem. (@CLAUDE, I'd bet that almost any book on measure theory would have it.) Sep 23, 2020 at 1:27

There is a very naive argument for this. As Henri says, it reduces to a zero set of a polynomial $f$. Write $$f(z_1,\ldots,z_n)=\sum_{j=0}^d g_j(z_1,\ldots,z_{n-1})z_n^j.$$ where the polynomial $g_d$ is not identically zero. For each $(z_1,\ldots,z_{n-1})\in\mathbb{C}^n$, there are only finitely many $z_n$ with $f(z_1,\ldots,z_n)=0$ unless $g_d(z_1,\ldots,z_{n-1})=0$. Inductively these exceptional $(n-1)$-tuples form a set of measure zero in $\mathbb{C}^{n-1}$ and now the result follows from Fubini's theorem (regarding $\mathbb{C}^n$ as $\mathbb{R}^{2n}$ and going down two real dimensions).

• Am I correct in saying that this argument works over any locally compact field (eg: zero sets of polynomials in $\mathbb{Q}_p^n$)? Jun 27, 2017 at 15:43
• with respect to the Haar measure... Jun 27, 2017 at 15:44
• Am I correct that this argument does not depend on the measure? In other words, does it show that Zariski-closed proper subsets of $\mathbb{C}^n$ are measure zero with respect to any measure?
– Ben
Mar 14, 2022 at 18:20

Clearly, it is sufficient too show it for a closed set given by $f=0$ where $f$ is analytic. (write your set as included in a countable union of such described sets). Then, using the normal form of analytic germs as finite ramified coverings, you're done.

And if you want to take this to an extreme... for a function on a domain in ${\bf R}^n$, it's enough to assume that at every $x$ there's a ball $B_x$ centered at $x$ and a multiindex $\alpha$ for which $\partial^{\alpha} f$ is nonzero and continuous on $B_x$.

To see this, first note that it suffices to show that the zeroes of $f$ in a given $B_x$ have measure zero. This is proven by induction on $|\alpha|$. If $\alpha = 0$ it's trivial, and if $\partial^{\alpha'}f(x) \neq 0$ for any $\alpha '$ with $|\alpha '| < |\alpha|$ it follows by the induction hypothesis, shrinking $B_x$ if necessary. Otherwise we can write $\partial^{\alpha} f = \partial_{x_i}\partial^{\beta} f$ for some $\beta$, where we can assume $\partial^{\beta} f(x) = 0$. By the implicit function theorem, if $B_x$ is small enough the zeroes of $\partial^{\beta} f$ in $B_x$ form a $C^1$ hypersurface with measure zero. For each $y$ off this surface, $\partial^{\beta} f$ is nonzero and then you can apply the inductive hypothesis to find an appropriate $B_y$. A simple compactness argument shows you need only countably many $B_y$.. so you're done.