In Wikipedia page of Dirac delta distribution (here), there is the generalization for a property of Dirac delta distribution given by
I know this is a generalization of the case when $g(\mathbf{x})$ has finite number of roots (in which case the right hand side becomes a summation over the roots). However, I have difficulty interpreting the integration measure. What is wrong with taking the right hand side to be simply the standard Riemannian integral over $f(\mathbf{x})/|\nabla g|$ over the level set $g^{-1}(0)$, i.e. over $d^{n-1}\mathbf{x}$, instead of what is called Minkowski content measure in the Wikipedia page?
I am looking for intuition and less so from rigorous interpretation in measure theory because I would like to use this for a very practical computation, even numerically perhaps. I would nonetheless appreciate some rigorous explanation on how to bridge the intuition and measure theory.