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Let $d\in\mathbb N$ and $f\in C^1(\mathbb R^d)$. Assume $\left\{\nabla f=0\right\}$ has Lebesgue measure $0$.

How can we conclude that $\left\{f\in B\right\}$ has Lebesgue measure $0$ for all Borel measurable $B\subseteq\mathbb R$ with Lebesgue measure $0$?

The claim can be found in an answer on mathoverflow.

The author writes that the claim "is true locally, in a neighborhood of each point where $\nabla f\ne0$, due to the implicit function theorem". Honestly, I don't even understand what exactly he's meaning.

Let $a\in\mathbb R^d$ with $\nabla f(a)\ne0$. Then surely, by continuity of $\nabla f$ at $a$, there is an open neighborhood $N$ of $a$ with $$\nabla f(x)\ne0\;\;\;\text{for all }x\in N\tag1.$$ But how do we need to apply the implicit function theorem and what's the resulting "local" conclusion? Maybe that $N\cap\left\{f\in B\right\}$ has Lebesgue measure $0$?

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    $\begingroup$ Yes you need to prove that $N\cap\{f\in B\}$ has measure $0$. If you show this the result follows, because there is a countable family of open sets $\{B_i\}$ covering $\Bbb R^d$ such that $B_i\cap\{f\in B\}$ has measure $0$. Then $\mu(\{f\in B\})=\sum_i \mu(\{f\in B\}\cap B_i)=0$. $\endgroup$ – Adam Chalumeau May 6 at 15:57
  • $\begingroup$ @AdamChalumeau Thank you for the clarification. I'm struggling to understand how we need to apply the implicit function theorem (I'm sure it's easy). $\endgroup$ – 0xbadf00d May 6 at 16:47
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Non-singular maps. A map $f\,:\,\mathbb{R}^N\rightarrow\mathbb{R}^M$ whose inverse image preserves null-sets, -- i.e., $\mu(f^{-1}(B))=0$ for any null-set $B$, -- is often referred to as a non-singular map. The question is about showing that the class of $\mathcal{C}^1$ maps with non-singular Jacobian (or more specifically, non-zero gradient) almost everywhere is contained in the class of non-singular maps.


Remark. $c\in\mathbb{R}$ is called a regular value of $f\in\mathcal{C}^{1}(\mathbb{R}^d)$, if $\nabla f(x)\neq 0$ for all $x\in f^{-1}(c)$. The Implicit Function Theorem (IFT) asserts that $f^{-1}(c)$ is a (d-1)-dimensional submanifold of class $\mathcal{C}^1$ -- for any regular value $c$. Hence, $f^{-1}(c)$ is a null-set.

Let $\widetilde{N}\overset{\Delta}=\left\{\nabla f\neq 0\right\}$ (which is open).

From the IFT, we have that $\widetilde{N}\cap f^{-1}(c)$ is a (d-1)-submanifold of class $\mathcal{C}^1$.

Now, you have $\widetilde{N}\cap \left\{f\in B\right\}=\bigcup_{t\in B}\widetilde{N} \cap f^{-1}(t)$, where $\widetilde{N} \cap f^{-1}(t)$ is a null-set for all $t$ from the above remark (since it is a $\mathcal{C}^1$ submanifold from the IFT). Therefore, when $B$ is countable, the referred set is a null-set.

When $B$ is uncountable, it follows from Fubini's Theorem that $\bigcup_{t\in B}\widetilde{N}\cap B_r \cap f^{-1}(t)$ is a null-set for any bounded open ball $B_r$.

To see this latter claim, we can resort to a more specialized form of Fubini tailored to our case (referred to as co-area formula),

$\int_{\widetilde{N}\cap B_r} g\left|\nabla f\right| d\mu = \int_{\mathbb{R}} \left(\int_{f^{-1}(t)\cap\widetilde{N}\cap B_r} g(x)d\mu_{d-1}(x)\right) dt$.

Take $g$ to be the indicator of the foliation $\bigcup_{t\in B}\widetilde{N}\cap B_r \cap f^{-1}(t)$ and note that

$\int_{\mathbb{R}} \left(\int_{f^{-1}(t)\cap\widetilde{N}\cap B_r} g(x)d\mu_{d-1}(x)\right) dt=\int_{B} \left(\int_{f^{-1}(t)\cap\widetilde{N}\cap B_r} d\mu_{d-1}(x)\right) dt=0$,

where the last identity holds since $B$ is a null-set. Thus,

$\int_{\widetilde{N}\cap B_r} g\left|\nabla f\right| d\mu=0$ and therefore $g\left|\nabla f\right|=0$ almost everywhere in $\widetilde{N}\cap B_r$. Since, $\left|\nabla f\right|\neq 0$ almost everywhere, it follows that $g(x)=0$ almost everywhere in $\widetilde{N}\cap B_r$. In other words,

$\mu\left(\bigcup_{t\in B}\widetilde{N}\cap B_r \cap f^{-1}(t)\right)=\int g d\mu =0$.


Update. For the sake of completeness, I am adding the general statement.

Theorem 1. Let $f\,:\,\mathbb{R}^N\rightarrow \mathbb{R}^M$ be smooth (i.e., $f\in\mathcal{C}^1$). If the set of critical points of $f$ is a null-set, i.e.,

$\mu\left(\left\{x\in\mathbb{R}^N : \text{rank} \left(Df(x)\right)<\min\left\{M,N\right\}\right\}\right)=0,$

then, $\mu\left(f^{-1}(B)\right)=0$ for any null-set $B$.

The proof follows from the IFT and Fubini (or, more precisely, the co-area formula) just as done before.


Update 2. I am adding a Corollary.

Definition. [Null-sets on manifolds] Let $\mathcal{V}$ be a smooth manifold of dimension $d$ with smooth structure $\left\{U_{\alpha},\varphi_{\alpha}\right\}$. $A\subset \mathcal{V}$ is called a null subset of $\mathcal{V}$ if $\mu\left(\varphi_{\alpha}(U_{\alpha}\cap A)\right)=0$ for all $\alpha$.

Relevant Property. If $\mu(\widehat{A})=0$ with $\widehat{A}\subset \mathbb{R}^d$ then,
$\varphi^{-1}_{\alpha}(\widehat{A})$ is a null-set for any $\alpha$. This follows by observing that $\varphi_{\beta}\left(U_{\beta}\cap\varphi^{-1}_{\alpha}(\widehat{A})\right)=\varphi_{\beta}\circ \varphi^{-1}_{\alpha}(\widehat{A})$ is necessarily a null-set, for any $\beta$, since $\widehat{A}$ is a null-set and $\varphi_{\beta}\circ \varphi^{-1}_{\alpha}$ is a diffeomorphism -- hence, from Theorem 1, $\varphi_{\beta}\circ \varphi^{-1}_{\alpha}(\widehat{A})$ is a null-set.

In the next corollary, we assume that the manifolds admit countable atlas -- i.e., are separable.

Corollary 1. Let $f\,:\,\mathcal{M}\rightarrow \mathcal{N}$ be a smooth map between two smooth separable manifolds $\mathcal{M}$, $\mathcal{N}$ of dimensions $M$ and $N$, respectively. If the set of critical points of $f$ is a null-set, then $f^{-1}(B)$ is a null-set for any null-set $B$.

For the proof, one just needs to notice that any local coordinate representation of $f$ fulfills the conditions of Theorem 1.

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  • $\begingroup$ It's a bit unfortunate that you redefined the symbol $N$ which is already used in the question. $\endgroup$ – 0xbadf00d May 6 at 18:29
  • $\begingroup$ Something is wrong in your post: (a) $f$ is a real-valued function. So, the notation $f^{-1}(c)$ (which I suppose is a shorthand for $f^{-1}(\{c\})$) is undefined. (b) Isn't a regular value defined to be the opposite of a critical/stationary point? (So, $a\in\mathbb R^d$ is a regular point of $f$ iff $\nabla f(a)\ne 0$. In this case, as mentioned in the question, there is a neighborhood $N$ of $a$ with $(1)$.) I guess you confused this with the notion of a regular value of $f$ which is precisely defined as the image of $f$ at a regular point. $\endgroup$ – 0xbadf00d May 6 at 18:49
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    $\begingroup$ If I am not missing anything, the $N$ you have in your question, is contained in the $N$ I have in my answer; (a) $f^{-1}(c)=\left\{x\in\mathbb{R}\,:\,f(x)=c \right\}$ refers to the inverse image of $c\in\mathbb{R}$ by $f$. Yes, one should rather write $f^{-1}(\left\{c\right\})$; (b) what is relevant is that $f^{-1}(c)$ is a surface whenever $c$ fulfills the above characterizing property (that I am referring in this case as regular). $\endgroup$ – Augusto Santos May 6 at 19:01
  • $\begingroup$ Yes, sure, but if I'm reading other questions I find it highly confusing if symbols are defined more than once. $\endgroup$ – 0xbadf00d May 6 at 19:06
  • $\begingroup$ Could you elaborate on how exactly we obtain that $f^{-1}(c)$ is a $(d-1)$-dimensional submanifold by the IFT? Since this is the only crucial part I don't understand (and actually the reason why I asked this question). $\endgroup$ – 0xbadf00d May 6 at 19:09
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I don't know about the implicit function theorem, but you can a one related theorem, the Local Submersion Theorem. With your notations, locally around $a$, $f$ looks like a projections onto the first coordinate. You are left to prove that $$p:(x_1,\dots,x_d)\in\Bbb{R}^d\mapsto x_1$$ has the property that $$(B\text{ has measure zero})\Longrightarrow (\{p\in B\}\text{ has measure zero}).$$ But $\{p\in B\}=B\times\Bbb R^{d-1}$ so you can conclude.

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  • $\begingroup$ You have to be careful with the degrees of smoothness: Usually, submersions are defined in the $C^\infty$ category, while OP is only assuming $C^1$. But Submersion theorem locally is the same as the Implicit Function Theorem. $\endgroup$ – Moishe Kohan May 6 at 19:25
  • $\begingroup$ I don't know the "local submersion theorem". However, the following is clear to me: If $M$ is a $k$-dimensional submanifold of $\mathbb R^d$ and $a\in M$, there is an open neighborhood $U_a$ of $a$ and a $C^1$-diffeomorphism $\varphi_a$ from $U_a$ to an open subset $V_a$ of $\mathbb R^d$ with $$\varphi_a(U_a\cap M)=V_a\cap(\mathbb R^k\times\{0\})$$ and hence $U_a\cap M$ is Borel measurable and has Lebesgue measure $0$ (assuming $k<n$). $\endgroup$ – 0xbadf00d May 7 at 15:50
  • $\begingroup$ Moreover, it's clear to me that if $c\in\mathbb R$ is a regular value of $f$ (i.e. $c=f(a)$ for some $a\in\mathbb R^d$ with $\nabla f(a)\ne 0$), then $M_c:=\{f=c\}$ is a $(d-1)$-dimensional submanifold of $\mathbb R^d$. But I don't know how we can conclude from that for the same reason I don't see why we can conclude in the other answer. It would be clear if $\{f\in B\}=\bigcup_{c\in B}M_c$ would be a countable union. $\endgroup$ – 0xbadf00d May 7 at 15:50

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