What is geometrical interptetation of a set being measurable.

Question

What is geometrical interptetation of a set being measurable. I mean what does it mean geometrically by a set is measurable...

Answer 1

On $\mathbb{R}^n$ with Lebesgue measure $\lambda$, non-measurable sets are pretty pathological, and their constructions depend on the Axiom of Choice, the axiom that allows you to split a ball into two identical ones.

Measurable sets are where $\lambda$ can act "properly". That is, any union of disjoint measurable sets has measure equal to the sum of individual measures.

The geometrical interpretation of a Lebesgue non-measurable set would be an object without a volume.

Answer 2

There is no geometric interpretation and there can be no nontrivial interpretation.

Nonmeasurable sets cannot arise through any geometrically meaningful operations, and it is consistent (with set theory minus uncountable Axiom of Choice, which includes all of concrete analysis and much more) to assume that they don't exist.

Therefore, any property that distinguishes measurable from a nonmeasurable sets has to be so ghostly that one can never find an example of a set without that property. In effect that means that the only distinctions are renamings of the property of being measurable in nearly identical words: whether the set

• has a measure
• has inner measure = outer measure
• has "volume", where volume is defined to mean measure.