How to prove that cuboids are Lebesgue measurable?

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• Maxi1995

Maxi1995

Hello,
how do I have to start to prove that cuboids are measurable in the context of the Lebesgue measure?
Best wishes
Maxi

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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.

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.
I think the subtlety comes when the cuboid is rotated.

The coordinate system can be rotated with it.

The coordinate system can be rotated with it.
Yes, but that is not part of the construction of the measure. One still needs to prove that it is invariant under rotations.

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