In mathematics, a G-measure is a measure μ that can be represented as the weak-∗ limit of a sequence of measurable functions G = ( G n ) n = 1 ∞ . A classic example is the Riesz product
G n ( t ) = ∏ k = 1 n ( 1 + r cos ( 2 π m k t ) ) where − 1 < r < 1 , m ∈ N . The weak-∗ limit of this product is a measure on the circle T , in the sense that for f ∈ C ( T ) :
∫ f d μ = lim n → ∞ ∫ f ( t ) ∏ k = 1 n ( 1 + r cos ( 2 π m k t ) ) d t = lim n → ∞ ∫ f ( t ) G n ( t ) d t where d t represents Haar measure. The
It was Keane who first showed that Riesz products can be regarded as strong mixing invariant measure under the shift operator S ( x ) = m x mod 1 . These were later generalized by Brown and Dooley to Riesz products of the form
∏ k = 1 ∞ ( 1 + r k cos ( 2 π m 1 m 2 ⋯ m k t ) ) where − 1 < r k < 1 , m k ∈ N , m k ≥ 3 .
