I mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer.
If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \geq N$ then $$ \int_A J_N f(x) d\mathcal{L}^M x = \int_{R^N} \mathcal{H}^{M-N} (A\cap f^{-1}(y)) d\mathcal{L}^N y $$
Could anyone please give me an intuition of what the formula implies and what it means practically? In particular, this is a simple example I am trying to wrap my head around: suppose we are in 2 dimensions and there is a curve defined by a function
What does the co-area formula tell us in this context?
Here are some practical questions:
- My understanding is that we have some space of dimension $M$ and there a manifold in it of dimension $M-N$ and we would like to compute the area of this manifold using the Hausdorff measure. Is this what is happening?
- I am completely lost as to why we need $A\cap f^{-1}(y))$ what's the intuition behind this?
- On the right hand side we have $\mathcal{H}^{M-N}$. I am familiar with the Lebesgue measure $d\lambda$ however I am lost as to why we seem to have both the Lebesgue and the Hausdorff measure on the right hand side. I thought we should only have the Hausdorff measure on the RHS?