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I still have no background in measure theory or Lebesgue integration. So is there any elementary-level resource --- book, set of notes, tutorial paper, website, or online video lecture series --- which I can use for getting some knowledge of these fields?

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  • $\begingroup$ Define "elementary level", please! $\endgroup$ – Professor Vector Jun 11 '17 at 9:01
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    $\begingroup$ Terence Tao's book Analysis II gives a good, relatively short introduction to measure theory and Lebesgue integration. After Tao's book, I recommend checking out Wheeden and Zygmund, or perhaps Stein and Shakarchi. $\endgroup$ – littleO Jun 11 '17 at 9:18
  • $\begingroup$ I found the part of Folland's "Real Analysis" dealing with measure theory and the Lebesgue integral explained concisely and clearly. I also personally enjoyed reading Billingsley's "Probability and Measure", but you have to beware that there is a lot of information in that book, and that it is big. You could maybe consult Folland's book first, and then, if you have some time and want to know more, consult Billingsley's book. $\endgroup$ – Malkoun Jun 11 '17 at 9:29
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I would like to suggest three books which helped me-

1) Royden's Real Analysis,here in this it gives motivation towards the topic as well as illustrative text,nice examples,excercises.

2)Measure Theory and Integration by G. de Barra.

3)Paul Halmos,Measure theory.

Also there are some similar questions asked and may contain some references as per your requirement - Reference book on measure theory and here.

And some online notes here and here.

Hope this helps!

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This is an introductory video course on measure and integration by Prof.Inder Rana of IIT Bombay. There are total 40 lectures with each one almost of an hour. Also he uses his own book on measure and integration along with Paul Halmos's and Royden's as mentioned in @BAYMAX's answer.

The syllabus of the course as well as the references are mentioned in the link.

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