# Probability distribution/measurment problem

The onedimensional measure problem can described by the following requirements: Is there an illustration with the following properties:

*Positivity*: for all*Translation Invariance*: for all and .*Normality*: ,- -
*Additivity*: if for ?

## Unsolvable of Measurement Problem

editThe onedimensional measurement problem cannot be solved on the power set on as shown by
Giuseppe Vitali in 1905, who was the first to give an example of a non-measurable subset of real numbers, see Vitali set.^{[1]} His covering theorem is a fundamental result in measure theory.

The onedimensional measurement problem can easily be extended to -dimensional spaces; this encompasses the dimensions 1, 2 and 3, that are accessible to our spatial perception, whereby

- dimension 1 refers to length,
- dimension 2 to an area in a plane and
- dimension 3 to measurment of the volume.

Due to the fact, that the measurement problem for cannot be solved as shown by Vitali, the sigma-algebra was introduced as the domain of the measure , that replaces especially the power set as domain of the probability measure on a -algebra .

## n-dimensional Case of the Measurement Problem

editThe measurement problem in the -dimensional case is:

Is there an measure with the following properties:

## References

edit- ↑ G. Vitali,
*Sul problema della misura dei gruppi di punti di una retta*, Bologna, Tip. Gamberini e Parmeggiani (1905).