Question
I am trying to get an intuition for the area formula as a "generalized" change of variables formula.
If $f:\mathbb{R}^m\to\mathbb{R}^n$ is Lipschitz and $m\leq n$ then $$ \int_A g(f(x)) J_m f(x) \lambda^m(dx) = \int_{\mathbb{R^n}} g(y) N(f \mid A, y) \mathcal{H}^m(dy) $$ whenever $A$ is $\lambda^m$-measurable, $g:\mathbb{R}^n\to \mathbb{R}$ is Borel and $N(f \mid A, y)=\#\left\{x\in A\, :\, f(x) = y\right\}$
My Attempt at an Intuition
The standard change of variables formula for integration by substitution is
Let $U\subset\mathbb{R}^n$ be an open set and $\phi: U\to\mathbb{R}^n$ an injective differentiable function with continuous partial derivatives, with non-zero Jacobian. Then for every real-valued, compactly-supported continuous function $f$ with support contained in $\phi(U)$ we have $$ \int_U f(\phi(\boldsymbol{u})) |\text{det}(D\phi)(\boldsymbol{u})|d \boldsymbol{u} = \int_{\phi(U)} f(\boldsymbol{v})d \boldsymbol{v} $$
One way in which I understand this formula is that if one is presented with the integral on the RHS and knows that $\boldsymbol{v} = \phi(\boldsymbol{u})$ then it is possible to write the integral in terms of $\boldsymbol{u}$ by:
- plugging in $\boldsymbol{v} = \phi(\boldsymbol{u})$
- mapping the domain of integration from $\phi(U)\subset V$ to $\phi^{-1}(\phi(U)) = U$, and
- correcting the integrand for the change in volume brought in by the transformation $|\text{det}(D\phi)(\boldsymbol{u})|$
Or similarly one can see it the other way around: if we find the integral on the LHS and we can recognise the second term to be the Jacobian of a transformation, then we can simplify the integral by substitution and get to the RHS.
What is a similar intuition for the Area formula above? Clearly, $f$ seems to be playing the role of $\phi$, our transformation, and the original function that we want to integrate is now called $g$. I can see that a major difference between these formulas is the fact that we have the Lebesgue measure on the LHS and the Hausdorff measure on the other. I am confused by the "cardinality" term $N(f \mid A, y)$ as this doesn't really appear in the usual change of variable formula.
