Probability distribution/measurment problem

The onedimensional measure problem can described by the following requirements: Is there an illustration $\mu _{n}:{\mathcal {P}}({\mathbb {R} }^{n})\ rightarrow[0,\infty ]$ with the following properties:

• Positivity: $\mu _{n}(A)\geq 0$ for all $A\subset {\mathbb {R} }$
• Translation Invariance: $\mu (A)\,=\,\mu (A_{v})$ for all $v\in {\mathbb {R} }$ and $A_{v}:=v+A:=\{v+a\,|\,a\in A\}$.
• Normality : $\ mu([0,1])\,=\,1$,
• $\sigma$-Additivity: $\mu (\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }\mu (A_{i})$ if $A_{i}\cap A_{j}=\emptyset$ for $i\not =j$?

Unsolvable of Measurement Problem

The onedimensional measurement problem cannot be solved on the power set on $\mathbb {R}$ 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. His covering theorem is a fundamental result in measure theory.

The onedimensional measurement problem can easily be extended to $n$-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 $\mathbb {R}$ cannot be solved as shown by Vitali, the sigma-algebra was introduced as the domain of the measure $\mu$, that replaces especially the power set as domain of the probability measure $P$ on a $\sigma$-algebra ${\mathcal {S}}$.

n-dimensional Case of the Measurement Problem

The measurement problem in the $n$-dimensional case is:

Is there an measure $\mu _{n}:{\mathcal {P}}({\mathbb {R} }^{n})\rightarrow [0,\infty ]$ with the following properties:

• Positivity: $\mu _{n}(A)\ ge0$ for all $A\subset {\mathbb {R} }^{n}$ (this condition is already in the default of the image set of the figure),
• Congruence: $\mu _{n}(A)\,=\,\mu _{n}(B)$ if A and B are congruent,
• Normality: $\ mu_{n}([0,1]^{n})\,=\,1$,
• $\sigma$-Additivity: $\mu _{n}(\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }\mu _{n}(A_{i})$ if $A_{i}\cap A_{j}=\emptyset$ for $i\not =j$?