51
$\begingroup$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is countable, the border of a set has measure zero, etc. Can you help me sharing your experience or with some reference list?

  • 10
    False belief: a set of positive measure $A$ contains an interval (but by Steinhaus theorem we know that it's the case for $A+A$); take fat Cantors.2011-12-24
  • 6
    False belief: if a function is continuous almost everwhere, then it is equal almost everywhere to a continuous functions, and vice versa.2011-12-24
  • 15
    @David: An easier counter example to "a set of positive measure $A$ contains an interval" is the irrationals. Fat Cantor sets are a better counter example to "a set of of positive measure $A$ is dense somewhere"2011-12-24
  • 6
    Counterexamples in Analysis - B. Gelbaum, J. Olmsted (Dover, 2003) would be a good reference here. e.g., they show: there is a measurable non-Borel set; there is a set of measure 0 that is not a countable union of closed sets;2011-12-24
  • 1
    This is not just about $\mathbb{R}$, but still: all $\mathbb{R}^n$'s (with the Lebesgue measure) are isomorphic as measure spaces; there is no "invariance of dimension" (as one might falsely believe)2011-12-24
  • 0
    I often find myself in situations where the Borel measure already measures every subset of $\mathbb R$. To say what sort of beliefs, one also have to specify the sort of axioms he assumes in the background.2011-12-24
  • 1
    I second David Mitra's advice to get a hold of Gelbaum & Olmsted's _Counterexamples in Analysis_.2011-12-24
  • 0
    False belief: Lebesgue is spelled with a q.2016-01-16
  • 0
    1.There might be something in Counter-Examples In Point-Set Topology. 2.Borel found an error in Lebesgue's famous monograph : The assertion that the projection of a 2-dimensonal real Borel set onto 1 co-ordinate is 1-dimensionally Lebesgue-measurable, which is not always true.This led to the discovery of a larger class : The projective sets.2016-01-16
  • 0
    @user254665 I believe you're thinking of [Suslin](https://en.wikipedia.org/wiki/Mikhail_Yakovlevich_Suslin) (not Borel) who caught a famous error of Lebesgue. He did not (of course) prove that the projection of a Borel set could be non-measurable, but he did show that it was not always a Borel set.2016-01-16
  • 0
    Right .I was going on memory. & it was Suslin too.2016-01-16

7 Answers 7