I have a mild curiosity as to why are we using the Hausdorff measure to define surface integrals (for example the co area formula) and not use instead the Lebesgue measure, or at least on my class and notes i have seen the definition like this:
Let $M\subset \mathbb{R}^{n}$ be a regular surface of dimension d, and g a function defined on that surface to the real numbers, g Hausdorff measurable, then we define the integral of g over M as:
$\int_{M}{g d\sigma}:=\int_{M}{g d\mathscr{H}^{d}}$
I am sorry if i don't really get my point across i am a bit confused on this.
I would appreciate any clarification!