G-measure

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

[math]\displaystyle{ G_n(t) = \prod_{k=1}^n \left( 1 + r \cos(2 \pi m^k t) \right) }[/math]

where [math]\displaystyle{ -1 \lt r \lt 1, m \in \mathbb N }[/math]. The weak-∗ limit of this product is a measure on the circle [math]\displaystyle{ \mathbb T }[/math], in the sense that for [math]\displaystyle{ f \in C(\mathbb T) }[/math]:

[math]\displaystyle{ \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 }[/math]

where [math]\displaystyle{ dt }[/math] 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 [math]\displaystyle{ S(x) = mx\, \bmod\, 1 }[/math]. These were later generalized by Brown and Dooley to Riesz products of the form

[math]\displaystyle{ \prod_{k=1}^\infty \left( 1 + r_k \cos(2 \pi m_1m_2\cdots m_k t) \right) }[/math]

where [math]\displaystyle{ -1 \lt r_k \lt 1, m_k \in \mathbb N, m_k \geq 3 }[/math].

External links

• Riesz Product at Encyclopedia of Mathematics