Geometry of measures in \(R^ n:\) Distribution, rectifiability, and densities.

*(English)*Zbl 0627.28008In his three fundamental papers of 1928, 1938 and 1939 A. S. Besicovitch explored the geometric measure theoretic structure of those subsets of the plane which are measurable and have finite measure with respect to the one-dimensional Hausdorff measure. He showed that any such set can be split into a regular and irregular part, where the geometric measure theoretic properties, e.g. tangential, projection, rectifiability and density properties, of the regular part are similar to those of rectifiable curves, whereas the irregular part has completely opposite behavior. In 1947 H. Federer developed the theory of k-dimensional rectifiable sets in \(R^ n\) and generalized most of Besicovitch’s results to arbitrary dimensions. The most important question that remained open in the higher-dimensional case was whether the existence of densities implies rectifiability. In the present paper this question is settled by the following theorem:

If \(0\leq m\leq n\) are integers and \(\phi\) is a Borel measure on \(R^ n\) such that, with \(B(x,r)=\{y: | x-y| \leq r\},\) \[ (1)\quad 0<\lim_{r\downarrow 0}\phi B(x,r)/r^ m<\infty \] for \(\phi\) almost all \(x\in R^ n\), then \(\phi\) almost all of \(R^ n\) can be covered by countably many m dimensional \(C^ 1\) submanifolds of \(R^ n.\)

In fact, the author proves more than this since rather than assuming that the limit actually exists, he shows that it is sufficient to assume that the ratio of the upper and lower limit is close to one. He also derives a great deal of information on the problem for which functions h (in place of \(r^ m)\) there exists a positive measure \(\phi\) such that \[ 0<\lim_{r\downarrow 0}\phi B(x,r)/h(r)<\infty \] for \(\phi\) almost all x. This extends a result of J. M. Marstrand from 1964 which tells that for \(h(r)=r^ m\) such a measure exists only if m is an integer.

The methods of this paper are very complicated and delicate. They utilize earlier ideas of Besicovitch, Federer and Marstrand, but the main ingredients consist of new algebraic techniques involving expansions and estimates for the moments \(\int e^{-s| z|^ 2}<z,x>^ kd\phi z.\)

To study measures satisfying (1), the author blows them up from small neighborhoods and takes weak limits. Such tangent measures \(\psi\) when suitably normalized satisfy \[ (3)\quad \psi B(x,r)=\alpha (m)r^ m\quad for\quad x\in \sup port \psi. \] Here \(\alpha\) (m) is the volume of the m dimensional unit ball. As an interesting result itself, he proves that in the cases \(m=0\), 1 or 2, (3) implies that \(\psi\) is the m dimensional Lebesgue measure on some m-plane. Surprisingly this is false for \(m\geq 3\).

If \(0\leq m\leq n\) are integers and \(\phi\) is a Borel measure on \(R^ n\) such that, with \(B(x,r)=\{y: | x-y| \leq r\},\) \[ (1)\quad 0<\lim_{r\downarrow 0}\phi B(x,r)/r^ m<\infty \] for \(\phi\) almost all \(x\in R^ n\), then \(\phi\) almost all of \(R^ n\) can be covered by countably many m dimensional \(C^ 1\) submanifolds of \(R^ n.\)

In fact, the author proves more than this since rather than assuming that the limit actually exists, he shows that it is sufficient to assume that the ratio of the upper and lower limit is close to one. He also derives a great deal of information on the problem for which functions h (in place of \(r^ m)\) there exists a positive measure \(\phi\) such that \[ 0<\lim_{r\downarrow 0}\phi B(x,r)/h(r)<\infty \] for \(\phi\) almost all x. This extends a result of J. M. Marstrand from 1964 which tells that for \(h(r)=r^ m\) such a measure exists only if m is an integer.

The methods of this paper are very complicated and delicate. They utilize earlier ideas of Besicovitch, Federer and Marstrand, but the main ingredients consist of new algebraic techniques involving expansions and estimates for the moments \(\int e^{-s| z|^ 2}<z,x>^ kd\phi z.\)

To study measures satisfying (1), the author blows them up from small neighborhoods and takes weak limits. Such tangent measures \(\psi\) when suitably normalized satisfy \[ (3)\quad \psi B(x,r)=\alpha (m)r^ m\quad for\quad x\in \sup port \psi. \] Here \(\alpha\) (m) is the volume of the m dimensional unit ball. As an interesting result itself, he proves that in the cases \(m=0\), 1 or 2, (3) implies that \(\psi\) is the m dimensional Lebesgue measure on some m-plane. Surprisingly this is false for \(m\geq 3\).

Reviewer: P.Mattila

##### MSC:

28A75 | Length, area, volume, other geometric measure theory |