This question comes from the proof of Lemma 9.3 of Bartle's "The Elements of Integration and Lebesgue Measure" in page 97-98. This proof is shown as the image below.(adsbygoogle = window.adsbygoogle || []).push({});

Form (9.1) mentioned in the lemma is: [tex](a,b], (-\infty,b], (a,+\infty), (-\infty,+\infty)[/tex].

My question is: although [tex]I_j[/tex] constructed in P98 is a bit fatter than [tex](a_j,b_j][/tex], I doubt the assertion that the left endpointa, and in turn the compact interval [a,b], is also covered by [tex]\{I_j\}[/tex], as the proof in the text claimed (I drew a red underline). Is my doubt correct (this means the text is incorrect), or pointacan be proved to be covered by [tex]\{I_j\}[/tex] (how)? Thanks!

PS: the establishment of the converse inequality does not need the coverage of the whole [a,b]. A small shrink, say [tex][a+\epsilon,b][/tex], is sufficient to get the inequality.

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# A question on proving countable additivity

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