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?
False beliefs about Lebesgue measure on $\mathbb{R}$
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measure-theory
examples-counterexamples
big-list
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10False 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
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6False belief: if a function is continuous almost everwhere, then it is equal almost everywhere to a continuous functions, and vice versa. – 2011-12-24
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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
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6Counterexamples 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
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1This 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
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0I 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
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1I second David Mitra's advice to get a hold of Gelbaum & Olmsted's _Counterexamples in Analysis_. – 2011-12-24
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0False belief: Lebesgue is spelled with a q. – 2016-01-16
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01.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
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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
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0Right .I was going on memory. & it was Suslin too. – 2016-01-16