Probability distribution/measurment problem

From Wikiversity
Jump to navigation Jump to search

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]

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