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If $\mathcal{O}$ denote the collection of real-valued functions defined on $[0,1]$ that map every set of measure zero to a set of measure 0. Prove that the set $\mathcal{O} $ is absolutely continuous. could anyone give me a hint for the proof?

Intuition
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  • This may be me being dense, but do you mean show that every function in $\mathcal{O}$ is absolutely continuous? – Keen-ameteur Dec 08 '19 at 12:27
  • yes I mean showing that every function in $\mathcal{O}$ ..... @Keen-ameteur – Intuition Dec 08 '19 at 12:33
  • It seems that what you wrote is that every function in $\mathcal{O}$ satisfies the Luzin property: https://en.wikipedia.org/wiki/Luzin_N_property. It says there that it is a necessary but not sufficient condition for absolute continuity. It also appears in the wikipedia page on absolute continuity. – Keen-ameteur Dec 08 '19 at 12:43
  • but it says also this "Also, if a function f on the interval [a,b] is continuous, is of bounded variation and has the Luzin N property, then it is absolutely continuous." @Keen-ameteur – Intuition Dec 08 '19 at 12:50
  • @Keen-ameteur are my functions continuous? are they of BV ? – Intuition Dec 08 '19 at 12:51
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    By what you wrote it seems that you defined $\mathcal{O}$, to be the set of real valued functions satisfying the Luzin property. So by what you quoted it seems that no. Also see this thread: https://math.stackexchange.com/questions/499101/example-of-a-function-that-has-the-luzin-n-property-and-is-not-absolutely-cont – Keen-ameteur Dec 08 '19 at 12:57
  • So you mean that this proof can not be done @Keen-ameteur .... correct? – Intuition Dec 08 '19 at 12:59
  • what about proving that every Lipschitz function belongs to the set mentioned above .....how can we prove this? (in the link of the question I provided above) – Intuition Dec 08 '19 at 13:00

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I'll write this here since we are starting to have a lengthy discussion in the comment section which we are not supposed to do.

As I said earlier, this page shows that the property you wrote is the Luzin property, and it also states that in the page on absolute continuity that it is part of the sufficient and necessary condition for absolute continuity:

$f: [a,b] → \mathbb{R}$ is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property.

This thread gives examples that it does not imply absolute continuity, and finally this thread shows how to show that Lipschitz functions are absolutely continuous.

To avoid spamming the comment section, I will edit this answer instead.

Keen-ameteur
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