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

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

Unsolvable of Measurement Problem

The onedimensional measurement problem cannot be solved on the power set on ${\displaystyle \mathbb {R} }$ 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.^[1] His covering theorem is a fundamental result in measure theory.

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

n-dimensional Case of the Measurement Problem

The measurement problem in the ${\displaystyle n}$-dimensional case is:

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

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

References

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