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Banach measure

Stefan Banach proved the existence of Banach measures in 1923.^[1] This established in particular that paradoxical decompositions as provided by the Banach-Tarski paradox in Euclidean space R³ cannot exist in the Euclidean plane R².

Definition[edit]

A Banach measure^[2]onRⁿ is a function ${\displaystyle \mu :{\mathcal {P}}(\mathbb {R} ^{n})\to [0,\infty ]}$ (assigning a non-negative extended real number to each subset of Rⁿ) such that

• μ is finitely additive, i.e. ${\displaystyle \mu (A\cup B)=\mu ($A)+\mu (B)} for any two disjoint sets ${\displaystyle A,B\subseteq \mathbb {R} ^{n}}$;
• μ extends the Lebesgue measure λ, i.e. ${\displaystyle \mu ($A)=\lambda (A)} for every Lebesgue-measurable set ${\displaystyle A\subseteq \mathbb {R} ^{n}}$;
• μ is invariant under isometriesofRⁿ , i.e. ${\displaystyle \mu ($A)=\mu (f(A))} for every ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ and every isometry ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$.

Properties[edit]

The finite additivity of μ implies that ${\displaystyle \mu (\varnothing )=0}$ and ${\displaystyle \mu (A_{1}\cup \cdots \cup A_{k})=\sum _{i=1}^{k}\mu (A_{i})}$ for any pairwise disjoint sets ${\displaystyle A_{1},\ldots ,A_{k}\subseteq \mathbb {R} ^{n}}$. We also have ${\displaystyle \mu ($A)\leq \mu (B)} whenever ${\displaystyle A\subseteq B\subseteq \mathbb {R} ^{n}}$.

Since μ extends Lebesgue measure, we know that ${\displaystyle \mu ($A)=0} whenever A is a finite or a countable set and that ${\displaystyle \mu ([a_{1},b_{1}]\times \cdots \times [a_{n},b_{n}])=(b_{1}-a_{1})\cdots (b_{n}-a_{n})}$ for any product of intervals ${\displaystyle [a_{1},b_{1}]\times \cdots \times [a_{1},b_{1}]\subseteq \mathbb {R} ^{n}}$.

Since μ is invariant under isometries, it is in particular invariant under rotations and translations.

Results[edit]

Stefan Banach showed that Banach measures exist on R¹ and on R². These results can be derived from the fact that the groups of isometries of R¹ and of R² are solvable.

The existence of these measures proves the impossibility of a Banach–Tarski paradox in one or two dimensions: it is not possible to decompose a one- or two-dimensional set of finite Lebesgue measure into finitely many sets that can be reassembled into a set with a different Lebesgue measure, because this would violate the properties of the Banach measure that extends the Lebesgue measure.^[3]

Conversely, the existence of the Banach-Tarski paradox in all dimensions n ≥ 3 shows that no Banach measure can exist in these dimensions.

AsVitali's paradox shows, Banach measures cannot be strengthened to countably additive ones: there exist subsets of Rⁿ that are not Lebesgue measurable, for all n ≥ 1.

Most of these results depend on some form of the axiom of choice. Using only the axioms of Zermelo-Fraenkel set theory without the axiom of choice, it is not possible to derive the Banach-Tarski paradox, nor it is possible to prove the existence of sets that are not Lebesgue-measurable (the latter claim depends on a fairly weak and widely believed assumption, namely that the existence of inaccessible cardinals is consistent). The existence of Banach measures on R¹ and on R² can also not be proven in the absence of the axiom of choice.^[4] In particular, no concrete formula for these Banach measures can be given.

References[edit]