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Transactions of the American Mathematical Society

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Algebraically invariant extensions of $ \sigma$-finite measures on Euclidean space

Author: Krzysztof Ciesielski
Journal: Trans. Amer. Math. Soc. 318 (1990), 261-273
MSC: Primary 28C10
MathSciNet review: 946422
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Abstract: Let $ G$ be a group of algebraic transformations of $ {{\mathbf{R}}^n}$, i,e., the group of functions generated by bijections of $ {{\mathbf{R}}^n}$ of the form $ ({f_1}, \ldots ,{f_n})$ where each $ {f_i}$ is a rational function with coefficients in $ {\mathbf{R}}$ in $ n$-variables. For a function $ \gamma :G \to (0,\infty )$ we say that a measure $ \mu $ on $ {{\mathbf{R}}^n}$ is $ \gamma $-invariant when $ \mu (g[A]) = \gamma (g)\cdot\mu (A)$ for every $ g \in G$ and every $ \mu $-measurable set $ A$. We will examine the question: "Does there exist a proper $ \gamma $-invariant extension of $ \mu ?$ We prove that if $ \mu $ is $ \sigma $-finite then such an extension exists whenever $ G$ contains an uncountable subset of rational functions $ H \subset {({\mathbf{R}}({X_1}, \ldots ,{X_n}))^n}$ such that $ \mu (\{ x:{h_1}(x) = {h_2}(x)\} ) = 0$ for all $ {h_1},{h_2} \in H,{h_1} \ne {h_2}$. In particular if $ G$ is any uncountable subgroup of affine transformations of $ {{\bf {R}}^n},\gamma (g{\text{)}}$ is the absolute value of the Jacobian of $ g \in G$ and $ \mu $ is a $ \gamma $-invariant extension of the $ n$-dimensional Lebesgue measure then $ \mu $ has a proper $ \gamma $-invariant extension. The conclusion remains true for any $ \sigma $-finite measure if $ G$ is a transitive group of isometries of $ {{\mathbf{R}}^n}$. An easy strengthening of this last corollary gives also an answer to a problem of Harazisvili.

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Keywords: Invariant $ \sigma $-finite measures, algebraic transformations of $ {{\mathbf{R}}^n}$, isometries of $ {{\mathbf{R}}^n}$
Article copyright: © Copyright 1990 American Mathematical Society