G-measure

In mathematics, a G-measure is a measure that can be represented as the weak-∗ limit of a sequence of measurable functions . A classic example is the Riesz product

where . The weak-∗ limit of this product is a measure on the circle , in the sense that for :

where represents Haar measure.

History

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It was Keane[1] who first showed that Riesz products can be regarded as strong mixing invariant measure under the shift operator . These were later generalized by Brown and Dooley [2] to Riesz products of the form

where .

References

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  1. ^ Keane, M. (1972). "Strongly mixing g-measures". Invent. Math. 16 (4): 309–324. doi:10.1007/bf01425715.
  2. ^ Brown, G.; Dooley, A. H. (1991). "Odometer actions on G-measures". Ergodic Theory and Dynamical Systems. 11 (2): 279–307. doi:10.1017/s0143385700006155.
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