G-measure explained

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)} ^{inf ty} ^{n =1} . A classic example is the Riesz productG _{n(t)} = \prod ^{n} _{k =1} \left( 1 + r \cos( 2 \pi m ^{k} t) \right)

where

-1 < r < 1 , m \in N . The weak-∗ limit of this product is a measure on the circle T , in the sense that for f \in C( T) :\int f d\mu = \lim _{n\toinfty} \int f(t) \prod ^{n} _{k =1} \left( 1 + r \cos( 2 \pi m ^{k} t)\right) dt = \lim _{n\toinfty} \int f(t) G _{n(t)} dt

where

dt represents Haar measure.

History

S(x) = mx \bmod 1 . These were later generalized by Brown and Dooley ^[2] to Riesz products of the form\prod ^{inf ty} _{k =1} \left( 1 + r _{k} \cos( 2 \pi m _{1m} _{2 …} m _{k} t) \right)

where

-1 < r _{k} < 1 , m _{k} \in N, m _{k} \geq 3 .

External links

• Riesz Product at Encyclopedia of Mathematics

Notes and References

1. Keane . M. . Strongly mixing g-measures . 1972 . Invent. Math. . 16 . 4 . 309–324 . 10.1007/bf01425715.
2. G. . Brown . A. H. . Dooley . Odometer actions on G-measures.. Ergodic Theory and Dynamical Systems . 11 . 2 . 1991 . 279–307 . 10.1017/s0143385700006155.