A local version of the Projection Theorem

Toby O'Neil

Appeared in The Proceedings of the LMS, July 1996.


We prove the following theorem:

Suppose that \(1\leq m\leq n\) are integers and \(\mu\) is a Borel measure on \(\mathbb{R}^n\) are such that for \(\mu\)-almost every \(x\),

Then \(\mu \) is \(m\)-rectifiable.


This is the main result from my PhD thesis and is the second paper I ever wrote (although it was the third published). It grew from the following result (which I started thinking about in the summer of 1993 whilst at a summer school in Finland and proved back in the UK in the autumn):

Suppose that \(\mu\) is a Borel measure on \(\mathbb{R}^2\) such that for \(\mu\)-almost every \(x\), Then \(\mu \) is \(1\)-rectifiable.

In particular, if all the tangent measures have supports that are lines or half-lines, then the original measure is \(1\)-rectifiable.

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Created:13 August 1996
Modified:15 August 2000