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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.
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 endpoint a, 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 point a can 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.
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 endpoint a, 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 point a can 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.