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.
A cuboid is a three-dimensional geometric shape with six rectangular faces. It is also known as a rectangular prism.
A cuboid is considered Lebesgue measurable if its volume can be accurately measured using the Lebesgue measure, which is a mathematical tool used to measure the sizes of sets in n-dimensional space.
To prove that a cuboid is Lebesgue measurable, one must show that the volume of the cuboid can be accurately measured using the Lebesgue measure. This can be done by dividing the cuboid into smaller measurable units and summing their volumes using the Lebesgue measure.
Lebesgue measurable cuboids have the property of being able to be divided into smaller measurable units, and their volume can be accurately measured using the Lebesgue measure. They also have a well-defined surface area, which can be calculated using the Lebesgue measure.
No, not all cuboids are Lebesgue measurable. A cuboid must have a well-defined volume and surface area in order to be considered Lebesgue measurable. If the cuboid is infinitely large or has a fractal surface, it may not be possible to accurately measure its volume or surface area using the Lebesgue measure.