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


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


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  ?


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