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  NOTES ON N.BOURBAKI’S Integration Notes by S. K. Berberian Notes on text through Ch.V,  § 5, No. 4 (10-19-08)  Sterling K. Berberian, Prof.Emer.The University of Texas at Austin(e-mail: berb@math.utexas.edu)Mathematics Subject Classification (2000): 28-01, 28Bxx, 43-xx, 46Exx  Foreword When I first studied the srcinal French fascicles of N. Bourbaki’s  Int´e-gration   (in the 1970’s) I made a handwritten working translation. WheneverI reached an assertion that I didn’t “see” after a few seconds of reflection(which was often), I opened a pair of braces  { }  and worked out an explana-tion to myself before proceeding further. This document is a transcriptionof those explanations, indexed by the page and line numbers of the text inthe published translation to which they pertain (N. Bourbaki,  Integration  ,Vols. I, II, Springer–Verlag, 2004). On restudying the work while preparingthe new translation (starting in 1998), I often consulted the notes, occa-sionally finding my explanations inadequate; the notes in their present formaspire to repair the inadequacies.All of the above is rather personal; what justification can there be forputting the notes out in front of everybody? Some of the gaps I thought Isaw proved to be trivial ( after   I saw the light); others took several days towork out—but there might have been shortcuts that I didn’t find. My mo-tivation was high; I was determined to acquire the prerequisites for readingthe author’s  Th´eories spectrales   and J. Dixmier’s books on operator alge-bras, so I ground at the obstacle until it yielded. In answer to the questionposed above: My hope is that the reader of   Integration  , who comes upona sticking point and does not have the leisure to grind it away, will find inthese notes an explanation that will enable him to keep going without lossof momentum.Paragraph 3 of the  mode d’emploi   (“To the Reader”) makes it plain thatthe typical reader of   Integration   comes to the task well-equipped. We caninfer that the “gaps” are there on purpose; the missing details were doubtlesspresent in an initial draft of a proof, but (if we may be permitted to readthe author’s mind) they have been deliberately pruned away to highlightthe main outline of the proof while leaving enough details to enable thedetermined reader to fill in the gaps. This is marvelous exercise for theresearch muscles, surely one of the author’s didactic aims. That is goodnews; the reader can be reassured that the gaps are well thought out andthe chances good that a moderate amount of effort will see him through andrender these notes superfluous.  Errata  for  Integration   (INT) III.52,   .  − 3. For I read L. IV.17,   . 2. After “belong to  A    ,” insert “are   f   ,”. IV.30,   .  − 2. Delete inverted comma at the end of the line. IV.40,   . − 15. For “step function” as translation of   fonction en escalier  ,see the Note for this line (on p. IV.x26 below) and especially the Note forIV.66,   .  − 15 , − 14 (on p. IV.x88). IV.49,  Footnote. The period at the end should immediately follow theparenthesis. IV.51,  Footnote. Vis-a-vis the translation of   fonction ´etag´ee  , see theNotes mentioned above in the entry for IV.40,   .  − 15. IV.73,   .  − 8. For “Cor.” read “Cor. 1”. IV.84,   .  − 13. For  | y − z |  δ   read  | y − z |  < δ  . (See the Note for thisline.) IV.88,  Running head. For  § 6 read  § 5. IV.108,   . 20. For TVS read EVT.  { See the Note for   . 20–24 onp. IV.x248 below. } IV.113,   . 1. The argument needed for (16 n ) is more complicated.  { Seethe Note for this line on p. IV.x265 below. } IV.115,   .  − 5. In the formula (19), for  f  ( x ) read  f  ( x ). IV.117,   . 21. For  h ( x ) = 1 read, say,  h ( z ) = 1. (The letter  x  isreserved for an element of M in the statement of the Corollary, and  y  occurslater in the proof as another element of M.) V.12,   .  − 9. For “No. 3” read “No. 4”. V.46,   .  − 10. For ( g α ) read ( g n ). V.49,   . 9. For “Lemma 1” read “Lemma 2”. V.77,   . 1 and  − 9. Lines incompletely printed (‘washed out’). VI.10,   . 9. For “ q   is gauge” read “ q   its gauge”. VI.14,   .  − 9. For “subset” read “subsets”.
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