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This is my first time trying to learn measure and i found there are several different definitions for measure which are not equivalent.(i.e. $\sigma$-ring, $\sigma$ field, $\sigma$ algebra etc)

I want to know not only the usual definition for measure but also differences between those definitions too! (i.e. Universal set)

Thank you in advance ..

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Measures are defined on $\sigma$-algebras. The notion of a $\sigma$-ring is useful for developing measure theory but it is not what measures are usually defined on. – Qiaochu Yuan Jan 4 at 21:39
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@Qiaochu Rudin PMA defined measure on $\sigma$ ring, so is it better to skip this chapter and study another book? If so, would you please recommend me a book which briefly but precisely defines measure? I just want to define measure, Lebesgue Integral and go back to Fourier Analysis – Katlus Jan 4 at 21:46
The last chapter of PMA is very terse. I recommend that you skip it and move to Rudin's Real & Complex Analysis or Folland's Real Analysis. See here for book recommendations: math.stackexchange.com/questions/46213/… – Ayman Hourieh Jan 4 at 21:55
@Katus: well, I don't see why you have to choose between one and the other. They're not mutually exclusive. – Qiaochu Yuan Jan 4 at 22:12
Also note that $\sigma$ algebra and $\sigma$ field are typically used interchangeably, even though (obviously) the naming algebraic structures are not the same. – gnometorule Jan 4 at 23:18

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