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These propositions are listed as examples in my script and I want to know the proofs (I am not able to prove them myself, unfortunately, because im not smart and very new to measure theory) :

  1. If $I=I_{1}\times ... I_n$ is a bounded Interval on $\mathbb{R}^n$, then I is measurable and it holds that $m(I)=|I_{1}|...|I_n|$ (m(I) is called the elementary n-dimensional volume

  2. Every unbounded Interval$\subset \mathbb{R}^n$ is measurable with $m(I) = \infty$ and $\mathbb{R}^n$ is measurable with $m(\mathbb{R}^n) = \infty$

  3. Every open Subset $G\subset \mathbb{R}^n$ is measurable.

  4. Every closed subset of $\mathbb{R}^n$ is measurable

  5. Every compact Subset of $\mathbb{R}^n$ has a finite measure

Do you know of a source where I can find proofs to these examples or what I have to search for to find them.

Thx in advance

share|cite|improve this question
Bartle, Elements of Integration. – Michael Greinecker Feb 3 '13 at 17:30
In these lecture notes: – malin Feb 3 '13 at 17:39
Zygmund, Royden, Stein... – leo Feb 5 '13 at 13:56

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