Let $U$ be an open set in $\mathbb{R}^n$, $T:U \rightarrow \mathbb{R}^n$ be a diffeomorphism defined on $U$, $\mu$ be the $d$-dimensional Hausdorff measure, $0<d<n$.
Is it true that we have the following formula:
For all borel measurable function $f:T(U)\rightarrow \mathbb{R}$, do we have $$ \int _{T(U)}f d\mu=\int_{U}f\circ T|det(DT)|^{d/n} d\mu $$ where $DT$ is the Jacobian of $T$?
No it isn't you cannot just "equally distribute" the determinant over the dimensions, as $T$s action will in general depend on the direction. Consider the following example: $U = \mathbb R^2$, $T = \binom{1\,0}{0\,2}$, $d=1$. Let $f = \chi_{[0,1]\times \{0\}}$. Then $$ \int_{\mathbb R^n} f \, d\mu = \mu([0,1]\times \{0\}) = 1 $$ but $$ \int_{\mathbb R^n} f \circ T (\det DT)^{1/2}\, d\mu = \sqrt 2 $$ as $T|_{[0,1]\times \{0\}} = \mathrm{id}$, $DT(x) = T$ for all $T$ and $\det T = 2$.