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  1. Rudin didn't explicitly define measure, so I turned to wiki and wonder if wiki's definition is widely accepted? enter image description here

https://en.wikipedia.org/wiki/Measure_(mathematics)#Definition

  1. I don't quite understand the infimum symbol in the expression of outer measure. If we take wiki's definition of $\mu$, then $\sum^n_{i=1} \mu(A_i)$ is monotonic increasing, right? Why should we add an inf to it when taking the limit?

Rudin's discussion when measure $\mu$ first appeared (page 303): enter image description here

enter image description here

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    $\begingroup$ As a footnote to the Answers you've received: You should also learn about inner measure $\mu^i.$ The inner measure $\mu^i(E)$ is the $\sup$ of $\mu (C)$ over all closed $C\subset E.$ A subset $E$ of $\Bbb R^n$ is $n$-dimensionally Lebesgue measurable iff $\mu^i(E)=\mu^*(E).$ $\endgroup$ – DanielWainfleet Nov 19 '17 at 5:06
  • $\begingroup$ Thanks for your note! I think this chapter is quite difficult to understand.@DanielWainfleet $\endgroup$ – Hank Nov 19 '17 at 22:12
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  1. Yes, this is the accepted definition. Rudin defines measure later, on page 310.

  2. The infimum is over all countable coverings of E by elementary sets. For example, if your elementary sets are rectangles, and E is the unit ball, then you are taking the infimum over all possible coverings the unit ball by rectangles (without the infimum, you could just cover the unit ball with a single rectangle $[-1,1]^n$, which has much larger measure (volume) than the ball.)

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Note that you have confused between "measure" and "outer measure".

The measure function $\mu$ is defined on the set of measurable sets which can be seen here as you have started with a $\sigma$ algebra for that.

On the other hand the outer measure is defined on any set. As the name suggests the "outer measure" is a measure of a set from outside and so whenever we are taking a cover of a set say $A\subset \Bbb R^n$ then we are taking the infimum of the lengths of members of that cover.

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