G-measure

In mathematics, a "G"-measure is a measure formula_1 that can be represented as the weak-∗ limit of a sequence of measurable functions formula_2. A classic example is the "Riesz product"
where formula_4. The weak-∗ limit of this product is a measure on the circle formula_5, in the sense that for formula_6:
where formula_8 represents Haar measure.
History.
It was Keane who first showed that Riesz products can be regarded as strong mixing invariant measure under the shift operator formula_9. These were later generalized by Brown and Dooley to Riesz products of the form
where formula_11.