G-measure

In mathematics, a G-measure is a measure $$\mu$$ that can be represented as the weak-∗ limit of a sequence of measurable functions $$G = \left(G_n\right)_{n=1}^\infty$$. A classic example is the Riesz product


 * $$ G_n(t) = \prod_{k=1}^n \left( 1 + r \cos(2 \pi m^k t) \right)$$

where $$-1 < r < 1, m \in \mathbb N$$. The weak-∗ limit of this product is a measure on the circle $$\mathbb T$$, in the sense that for $$ f \in C(\mathbb T)$$:


 * $$\int f \, d\mu = \lim_{n\to\infty} \int f(t) \prod_{k=1}^n \left( 1 + r \cos(2 \pi m^k t)\right) \, dt = \lim_{n\to\infty} \int f(t) G_n(t) \, dt $$

where $$dt$$ 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 $$S(x) = mx\, \bmod\, 1$$. These were later generalized by Brown and Dooley to Riesz products of the form


 * $$ \prod_{k=1}^\infty \left( 1 + r_k \cos(2 \pi m_1m_2\cdots m_k t) \right)$$

where $$-1 < r_k < 1, m_k \in \mathbb N, m_k \geq 3$$.