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A How to prove that cuboids are Lebesgue measurable?

  1. Sep 15, 2018 at 5:09 AM #1
    Hello,
    how do I have to start to prove that cuboids are measurable in the context of the Lebesgue measure?
    Best wishes
    Maxi
     
    Last edited: Sep 15, 2018 at 5:54 AM
  2. jcsd
  3. Sep 15, 2018 at 4:09 PM #2

    mathman

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    Measure theory starts from the definition of measure of an interval is equal to its length. A cuboid is just the 3d analog of an interval. You can start with measure equals volume or you can define 3 space as the direct product of 3 lines and go on from there.
     
  4. Sep 16, 2018 at 12:54 AM #3

    martinbn

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    I think the subtlety comes when the cuboid is rotated.
     
  5. Sep 16, 2018 at 4:41 PM #4

    mathman

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    The coordinate system can be rotated with it.
     
  6. Sep 17, 2018 at 10:23 AM #5

    martinbn

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    Yes, but that is not part of the construction of the measure. One still needs to prove that it is invariant under rotations.
     
  7. Sep 17, 2018 at 4:15 PM #6

    mathman

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    Gooogle "Lebesgue measure lecture notes" - this will get you more answers.
     
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