G-measure

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

${\displaystyle G_{n}(t)=\prod _{k=1}^{n}\left(1+r\cos(2\pi m^{k}t)\right)}$

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

${\displaystyle \int f\,d\mu =\lim _{n\to \infty }\int f(t)\prod _{k=1}^{n}\left(1+r\cos(2\pi m^{k}t)\right)\,dt=\lim _{n\to \infty }\int f(t)G_{n}(t)\,dt}$

where ${\displaystyle dt}$ represents Haar measure.

History

It was Keane^[1] who first showed that Riesz products can be regarded as strong mixing invariant measure under the shift operator ${\displaystyle S(x)=mx\,{\bmod {\,}}1}$. These were later generalized by Brown and Dooley ^[2] to Riesz products of the form

${\displaystyle \prod _{k=1}^{\infty }\left(1+r_{k}\cos(2\pi m_{1}m_{2}\cdots m_{k}t)\right)}$

where ${\displaystyle -1<r_{k}<1,m_{k}\in \mathbb {N} ,m_{k}\geq 3}$.