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I may be mistaken but I understood that Borel developed his sigma algebra before Lebesgue developed his measure. If correct, then Borel can't have been trying to find a collection of sets in $\mathbb{R}$ that would support his student's measure. What was his motivation? Maybe my premise is wrong.

I've Googled and checked this exchange (and other similar exchanges) and I've not been able to get even a hint that this question has been asked.

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    $\begingroup$ See Thomas Hawkins, Lebesgue's theory of integration (AMS, 1975), Ch.4.2 Borel's Theory of Measure, $\endgroup$
    – Mauro ALLEGRANZA
    Apr 29, 2022 at 6:37
  • $\begingroup$ @MauroALLEGRANZA: Thanks for refs $\endgroup$
    – TonyK
    Apr 29, 2022 at 15:50
  • $\begingroup$ You are welcome :-) $\endgroup$
    – Mauro ALLEGRANZA
    Apr 29, 2022 at 16:11

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Borel was trying to understand "countable probabilities". For example, problems like this: for a given sequence $a_n\to 0, a_n>0$, what is the probability that the series $\sum\pm a_n$ converges? Or the "StPetersburg game": you play against a casino. A fair coin is tossed repeatedly, until the first appearance of "eagle". If this happens on $n$-th toss, you win $2^n$. What is the fair entrance fee for this game?

He explained his approach himself in his nice little book Probability and certainty.

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  • $\begingroup$ Typo - PeterSburg and I think we normally call it the St Petersburg game/paradox/and so on. $\endgroup$
    – mdewey
    Apr 29, 2022 at 12:59
  • $\begingroup$ Thanks Alexandre. That about does it. $\endgroup$
    – TonyK
    Apr 29, 2022 at 15:53

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