Summary
We consider a one-to-one, bi-measurable, non-singular transformation Φ of a finite measure space onto itself. We obtain two conditions which are equivalent to the existence of a σ-finite measure Μ which is invariant with respect to Φ and equivalent to the given measure m. The first is a generalization of a condition used by Ornstein in his construction of a transformation for which there does not exist any measure Μ as above. The second condition asserts that the entire space is the union of a countable collection {F¦ of subsets, each of which has the following property: if we countably decompose F in such a way that each set in the decomposition of F has an image (under some power of Φ) which is also a subset of F, then the sum of the m-measures of the images is finite (even though the images need not be disjoint).
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References
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Arnold, L.K. A note on σ-finite invariant measures. Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 93–95 (1969). https://doi.org/10.1007/BF00537515
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DOI: https://doi.org/10.1007/BF00537515