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I would like to ask you how to find a continuous function f so that for a Lebesgue-zero-set N we get λ(f(N)) > 0 wit λ being the Lebesgue measure.

Any chance I can work with the Cantor function? But if so, how would that work?

Thank for answers in advance, I would really appreciate your help. :)

Best KingDingeling

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  • $\begingroup$ Indeed something with the Cantor set helps you, since it is uncountable and has measure zero. This might help you math.stackexchange.com/q/40504/349501 $\endgroup$
    – Shashi
    Commented Dec 1, 2018 at 23:25
  • $\begingroup$ This link definitely helped, thank you very much. $\endgroup$
    – KingDingeling
    Commented Dec 2, 2018 at 12:30

2 Answers 2

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The Cantor function maps $[0,1]$ onto itself and it is constant on the intervals removed in the construction of the Cantor set $C$. Hence the image of $C$ under this function is $[0,1]$ minus a countable set so the image has measure $1$.

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  • $\begingroup$ Thank you for your help :), appreciate it. $\endgroup$
    – KingDingeling
    Commented Dec 2, 2018 at 12:30
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The Cantor function has the property that it maps $C$ onto $[0,1]$. So it definitely works. Depending on the exact definition we could have that it maps $C$ onto $[0,1]$ minus a countable set, which still has full measure $1$, so it's irrelevant.

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  • $\begingroup$ Thank you for your help :) $\endgroup$
    – KingDingeling
    Commented Dec 2, 2018 at 12:30

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