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

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

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

External links

• Riesz Product at Encyclopedia of Mathematics