Jan Maly: Non-absolutely convergent integrals with respect to distributions. Preprint MATH-KMA-2011/374, Charles University, Praha 2011 The preprint series
Jan Maly:
Lectures on change of variables in integral
Mathematical Analysis and Logic: Graduate School in Helsinki
The formula for change of variables in integral
$$
\int_{\Omega}v(f(x))|J_f(x)|\,dx = \int_{f(\Omega)}v(y)\,dy
$$
well known from the courses of calculus becomes more complicated if
the mapping $f$ is not one-to one, or the dimensions for the variables
$x\in \er^n$ and $y\in \er^d$ are not the same. It is interesting
to observe what quality of $f$ is needed for validity of the formula.
The purpose of the lecture is to prove the advanced formula on change
of variables distinguishing the cases $d\ge n$ (area formula) and $d\le n$
(coarea formula).
In order to set these results in a reasonable generality,
some background material from real analysis, Hausdorff measures and
Sobolev spaces is prepared. The class of generalized Lipschitz functions in
the sense of Rado and Reichelderfer as a natural class connected with the topic
is investigated.