Английская Википедия:G-measure

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

<math> G_n(t) = \prod_{k=1}^n \left( 1 + r \cos(2 \pi m^k t) \right)</math>

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

<math>\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 </math>

where <math>dt</math> 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 <math>S(x) = mx\, \bmod\, 1</math>. These were later generalized by Brown and Dooley to Riesz products of the form

<math> \prod_{k=1}^\infty \left( 1 + r_k \cos(2 \pi m_1m_2\cdots m_k t) \right)</math>

where <math>-1 < r_k < 1, m_k \in \mathbb N, m_k \geq 3</math>.

External links

• Riesz Product at Encyclopedia of Mathematics