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I recently stumbled upon the concept of Lebesgue measure while studying the construction of the Riemann integral and the Henstock–Kurzweil integral (also known as the gauge integral). As an undergraduate student of mathematics who has not had a course in measure theory, theorems related to the work of Henri Lebesgue seem quite intimidating. However, after reading a proof of Lebesgue’s Criterion for Riemann integrability I realized that the proof was quite simple and intuitively it makes a lot of sense to me. I've got a couple of question that I still have not found an answer to.

$1.$ In the definition of the Lebesgue outer measure, there is mention of intervals $I_1,I_2,I_3,...$ such that $\bigcup I_k$ contains the interval we want to measure. Why do all these $I_k$ have to be open?

$2.$ Can all Henstock–Kurzweil integrable functions be categorized by some set of functions that have a certain Lebesgue measure?

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    $\begingroup$ 1. they don't. it may make certain parts of the theory easier to require it, but its equivalent to using any kind of interval $\endgroup$
    – Tim kinsella
    Oct 27, 2018 at 20:20

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For the definition of the lebesgue outer measure the intervals don't have to be open. You can replace them e.g. by closed intervals or half-open intervals and get the same outer measure. Moreover the outer measure is defined for all sets, not only for intervals as your question suggests.

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