Did Grothendieck reinvent the Lebesgue integral?

Question

I have heard multiple times the story that while being a first year undergraduate student, mathematician Alexander Grothendieck rediscovered the Lebesgue integration, which was not so known at this time.

Is this true?

Answer 1

Biographer and fellow mathematician Winfried Scharlau wrote an article "Who is Alexander Grothendiek" [Originally published in German as “Wer ist Alexander Grothendieck?” in the Annual Report 2006 of the Mathematics Research Institute.]

While describing Grothendiek's early studies, he says:

Since the time he was in school, he had planned to find out what concepts like length and volume really mean, and according to his own reports, he basically developed the theory of the Lebesgue integral.

There is no direct citation for this claim.

Here we have a biographer claiming the subject claimed it directly. This, I suspect, is as close as we are likely to get to confirmation of the story.