Example of a Lebesgue measurable function which is not a Baire function?

Question

I found the following statement on Wikipedia :

"Henri Lebesgue proved that (for functions on the unit interval) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class."

I know that any non-measurable function can't be a member of any Baire class. But a proof of this kind requires the axiom of choice (for existence of a non-measurable set).

Did Lebesgue use AC in his proof? If not can anyone provide an example of a measurable function that is not in any Baire class?

(Or was Lebesgue's proof not constructive?)

ADDED: On Wolfram I found the following statement:

Lebesgue showed that each of the Baire classes is nonempty and that there are (Lebesgue-) measurable functions that are not Baire functions (Kleiner 1989).

So now the question winds down to: Can anyone give a sketch of a proof for this? or better yet an example of such a function?

Here's the paper, published in 1905, by Lebesgue: Sur les fonctions représentables analytiquement. You may want to try a google-phrase search for its title to see what turns up. — Dave L. Renfro, May 21 '14 at 21:50

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

1

Turns out it's simpler than i previously thought.

The Borel functions coincide with the Baire functions (on suitable topological spaces).

So any indicator function of a Lebesgue measurable set that is not Borel measurable will do.

edited Apr 13 '17 at 12:19

Community

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answered May 23 '14 at 08:42

Saal Hardali

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Example of a Lebesgue measurable function which is not a Baire function?

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