Cuboids are Lebesgue measurable as they are the three-dimensional analogs of intervals, with their measure defined as volume. The foundational aspect of measure theory begins with the definition of the measure of an interval equating to its length. To establish the measurability of cuboids, one can either define three-dimensional space as the direct product of three lines or start from the volume definition. It is essential to demonstrate that the Lebesgue measure remains invariant under rotations of the cuboid.
PREREQUISITESMathematicians, students of measure theory, and anyone interested in the properties of geometric shapes in relation to Lebesgue measurability.
I think the subtlety comes when the cuboid is rotated.mathman said: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.
Yes, but that is not part of the construction of the measure. One still needs to prove that it is invariant under rotations.mathman said:The coordinate system can be rotated with it.