In mathematics, a G-measure is a measure \( \mu \) 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 \( \mathbb T \), 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 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 \( {\displaystyle -1<r_{k}<1,m_{k}\in \mathbb {N} ,m_{k}\geq 3} \).
References
Keane, M. (1972). "Strongly mixing g-measures". Invent. Math. 16 (4): 309–324. doi:10.1007/bf01425715.
Brown, G.; Dooley, A. H. (1991). "Odometer actions on G-measures". Ergodic Theory and Dynamical Systems. 11 (2): 279–307. doi:10.1017/s0143385700006155.
External links
Riesz Product at Encyclopedia of Mathematics
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
