# Aronszajn measure

In a geometric measure theory (GMT) course I'm following this year, the professor told us about the Aronszajn measure, and asked us to go check by ourselves what it reprensents (the course was about measure theory in infinite dimensional Banach space). Unfortunalety, I didn't find on the internet what I was looking for, so I'd like to get some informations about these measures (including the definition), and references if possible.

Edit : Aronszajn measure are not defined, but there is a notion of negligeable sets. The definition is in the article mentionned by M. Peters or Handbook of the Geometry of Banach Spaces Volume 2 in the section of David Preiss. To be complete, I mention brielfly the definition. if $X$ is an infinite dimensional separable Banach space, and $(a_n)_{n\in\mathbb{N}}$ a dense sequence, a Borel subset $E\subset X$ is an Aronzajn null set if $E=\bigcup_{n\in \mathbb{N}}E_n$, $E_n$ is Borel, and for all $x\in X$, $\mathscr{H}^1(E_n\cap(x+\mathbb{R}a_n))=0$, where $\mathscr{H}^1$ is the one-dimenional Hausdorff measure.

• Curious: what does GMT stand for? – Nate Eldredge Nov 5 '14 at 15:17
• Google's only hits for the phrase "Aronszajn measure" (in quotes) are to this post. It doesn't seem that this term is generally used. Maybe you ought to ask the professor for clarification. – Nate Eldredge Nov 5 '14 at 15:19
• @NateEldredge, presumably GMT stands for "geometric measure theory" – Otis Chodosh Nov 5 '14 at 15:22
• @NateEldredge Excuse me, of course GMT stands for geometric measure theory, I thought that it was clear. – Paul-Benjamin Nov 5 '14 at 15:30
• «Of course»? Well, it could have meant Greenwich Mean Time and all sort of other things. It is never a good idea to have the third word of anything one writes be an accronym unless the audience is very, very limited. In any case, what does make me curious is if you asked this question to your professor right after googling and finding nothing! – Mariano Suárez-Álvarez Nov 7 '14 at 22:07

## 1 Answer

A look at mathnet.ru yields one paper by Bogachev on this topic. The article by Csörnyei in Israel Journal of Mathematics, December 1999, Volume 111, Issue 1, pp 191-201 and the references therein give further information.

• Thank you M. Peters, I found that the so-called Aronszajn measure doesn't exist, but negligeable sets are well defined. – Paul-Benjamin Nov 7 '14 at 21:51