False beliefs about Lebesgue measure on R\mathbb{R}

I’m trying to develop intuition about Lebesgue measure on 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?

Answer

False belief: the continuous image of a measurable set is measurable.

A counterexample is provided by the Devil’s staircase. Since the image of the Cantor set has full measure, it will have subsets, still measurable, which have non-measurable image. The same function also serves as a counterexample to the following:

False belief: if a continuous function has derivative zero almost everywhere, then it is constant.

Attribution
Source : Link , Question Author : Community , Answer Author : JDH

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