# Category Archives: Classic results

## Ergodic Decomposition

— 1. Introduction — In the study of measurable dynamics, the basic object of study is a measure preserving system: a quadruple , where is a set, is a -algebra over , is a probability measure on and is a … Continue reading

Posted in Analysis, Classic results, Ergodic Theory, Tool | | 22 Comments

## Disintegration of measures

In this post I will talk about conditional expectation and disintegration of a measure with respect to a -algebra. All this is classical probability theory but I think not many people (me included) come across this in a standard course … Continue reading

Posted in Analysis, Classic results, Tool | | 7 Comments

## The Ergodic Theorem

— 1. Introduction — One can argue that (modern) ergodic theory started with the ergodic Theorem in the early 30’s. Vaguely speaking the ergodic theorem asserts that in an ergodic dynamical system (essentially a system where “everything” moves around) the … Continue reading

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## Equidistribution of polynomials, recurrence and van der Corput trick

In the early twenty century Hardy and Littlewood gave an answer to the following problem: given a polynomial with real coefficients, when do we have that the set is dense in the interval ?. Today we have a much shorter … Continue reading

Posted in Classic results, Ergodic Theory | 9 Comments

## Polya’s criterion for positive definite sequences.

1. Introduction Let be the Torus. A function can be described by it’s Fourier series. We can, more generally, consider any Borel complex measure . The Fourier coefficients then look like , where, as usual, denotes the character of associated … Continue reading