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 $G={\left(G_n\right)}_{n=1}^{\mathrm{\infty }}$. A classic example is the Riesz product

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

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

$\int fd\mu =\underset{n\to \mathrm{\infty }}{\lim }\int f(t)\prod _{k=1}^n(1+r\cos (2\pi m^{k}t))dt=\underset{n\to \mathrm{\infty }}{\lim }\int f(t)G_n(t)dt$

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 $S(x)=mxmod1$. These were later generalized by Brown and Dooley to Riesz products of the form

$\prod _{k=1}^{\mathrm{\infty }}(1+r_k\cos (2\pi m_1{m}_2\cdots m_{k}t))$

where $-1<r_k<1,m_k\in \mathbb{\mathbb{N}},m_k\ge 3$.

External links

• Riesz Product at Encyclopedia of Mathematics