# A How to prove that cuboids are Lebesgue measurable?

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1. Sep 15, 2018 at 5:09 AM

### Maxi1995

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. Sep 15, 2018 at 4:09 PM

### mathman

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.

3. Sep 16, 2018 at 12:54 AM

### martinbn

I think the subtlety comes when the cuboid is rotated.

4. Sep 16, 2018 at 4:41 PM

### mathman

The coordinate system can be rotated with it.

5. Sep 17, 2018 at 10:23 AM

### martinbn

Yes, but that is not part of the construction of the measure. One still needs to prove that it is invariant under rotations.

6. Sep 17, 2018 at 4:15 PM

### mathman

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