I am trying to understand the following formula.
Let $f: A \subset \mathbb{R}^n \to \mathbb{R}^m$ be Lipschitz, with Lipschitz constant Lip $f$, $1 \leq n \leq m$. Then we have
$$ \int_{\mathbb{R}^m} H^0(f^{-1}(y) \cap A) dH^n(y) \leq (Lip \; f)^n L^n(A) $$
Where $H^0$ is the counting measure, and $f^{-1}(y) = \{ x : f(x) = y \}$.
Let $f: A = [-1,1] \subset \mathbb{R} \to \mathbb{R}^2 $ be defined by $f(t) = (|t|, |t|)$. Then $f$ is Lipschitz with Lip $f =1$. EDIT: My error as pointed out by ArcticChar in the comments, the lipschitz constant is $\sqrt{2}$ so it all works out.
We have $H^0(f^{-1}(y) \cap A) = 0 $ except on $f(A)/ \{0\}$ where it's 2.
Then,
$$ \int_{\mathbb{R}^2} H^0(f^{-1}(y) \cap A) dH^1(y) = 2 H^1(f(A)) = 2 \sqrt{2} $$
But the RHS of the inequality is $L^1([-1,1]) = 2$. What is wrong with my computation? EDIT: The error was earlier