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I am looking for a very gentle first book on measure theory. I want to acquaint myself with the basic ideas in Measure theory. Briefly, my background is as follows.

I am self-learning basic probability theory through a couple of books:

  • Introduction to Probability theory and its applications, Volume I - William Feller. (Currently reading Chapter X. Law of Large Numbers).

  • Probability and Random Processes - Grimmett and Stirzaker (Currently studying Chapter II. Random variables and distribution functions).

I am also learning Real Analysis (at the level of Baby Rudin) through the book:

  • Understanding Analysis, Stephen Abbott. (Currently reading chapter II on sequences and series).

One of the books I have in mind, is Sheldon Axler's book on measure theory (it's modern). Do you think I can jump into it anyway, given my current background? Also, any video playlists(lectures) are welcome.

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    $\begingroup$ By the way, the book by Feller that you’re learning from is excellent $\endgroup$ Dec 18, 2020 at 17:29
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    $\begingroup$ With your current background in real analysis Axler’s text is definitely out of scope. $\endgroup$ Dec 18, 2020 at 17:33
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    $\begingroup$ @Quasar I'm not really familiar with Axler's book, but Understanding Analysis isn't very comprehensive. If you don't want to use Rudin (which no sane person should in my opinion), I suggest having a look at Terence Tao's two-volume Analysis to fill any gaps in your knowledge (I've certainly quite liked it). He even goes into the basics of measure theory in the second volume. $\endgroup$ Dec 18, 2020 at 18:00
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    $\begingroup$ You can always dive in to a math book you're not ready for and then backtrack to fill in gaps in knowledge as necessary. I'd guess that Axler's book is definitely worth spending some time on, even if you ultimately choose to focus more on a different book. $\endgroup$
    – littleO
    Dec 18, 2020 at 21:13
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    $\begingroup$ My book Measure, Integration & Real Analysis (the electronic version is legally available for free at measure.axler.net) should be accessible after you finish Abbott's book. Also, the website for my book includes a free Supplement that reviews some of the background material. $\endgroup$ Dec 19, 2020 at 2:26

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Check out A First Look at Rigorous Probability Theory by Rosenthal. The first five chapters, in particular, seem like exactly what you're looking for: a very gentle introduction to the basics of measure theory with an eye toward probability.

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    $\begingroup$ i was about to say this: i suggest, like aduh, A First Look at Rigorous Probability Theory by Rosenthal. i further suggest that probability with martingales by david williams be NOT used as a main text, but i suggest williams' book as a supplement. $\endgroup$
    – BCLC
    Dec 19, 2020 at 2:04

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