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Dec 18, 2010 at 2:15 vote accept ronggang
Dec 17, 2010 at 15:40 comment added Andrés E. Caicedo On the other hand, if one assumes determinacy (that contradicts choice), then the result is true (at least in any decent space). For example, for the reals we in fact have that if the only Borel subsets of $A$ are measure zero, then $A$ is contained in a Borel set of measure zero. Note that Joel's argument does not use choice after the first paragraph.
Dec 17, 2010 at 5:19 history edited Joel David Hamkins CC BY-SA 2.5
fixed error regarding finiteness of measure
Dec 17, 2010 at 4:57 history answered Joel David Hamkins CC BY-SA 2.5