# 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[edit | edit source]

The 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[edit | edit source]

The measurement problem in the -dimensional case is:

Is there an measure with the following properties:

**Positivity**: for all (this condition is already in the default of the image set of the figure),**Congruence**: if*A*and*B*are congruent,**Normality**: ,- -
**Additivity**: if for ?

## References[edit | edit source]

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