Is every Baire one function define on $\mathbb{R}$ continuous a.e.?

Asked Mar 30 '17 at 07:19

Active Mar 30 '17 at 07:19

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Is every Baire one function continuous a.e.? I guess it has a positive answer. Because

$f:\mathbb{R} \to \mathbb{R}$ is a Baire one function iff $f$ continuous everywhere except for meagre set.
A subset $A$ of $X$ is meagre that it is negligible. So, we can assume that $A$ has a measure zero (I am not sure in this part).

Therefore, $f:\mathbb{R} \to \mathbb{R}$ is a Baire one function iff the discontinuity set of $f$ has measure zero.

asked Mar 30 '17 at 07:19

flourence

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Not every meagre set has measure zero, and you can't just "assume" it is. – Wojowu Mar 30 '17 at 07:20
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http://math.stackexchange.com/questions/102482/how-badly-can-dinis-theorem-fail-if-the-p-w-limit-isnt-continuous – Jonas Meyer Mar 30 '17 at 07:27
@JonasMeyer. Thanks for the link and the answer – flourence Mar 30 '17 at 07:46
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Note that @Jonas Meyer shows that a Baire one function on $\mathbb R$ can be discontinuous a.e.! – Dave L. Renfro Mar 30 '17 at 14:13
Some types of meager closed sets of positive measure are called "fat Cantor sets". – DanielWainfleet Mar 30 '17 at 21:48

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