# 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

## 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

## 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

## 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

## 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

## On different ways to take limits

One of the most fundamental notions in modern mathematics is the notion of limit. And to define limit of a sequence, the only structure needed is a Hausdorff topology (if we want the limit to be unique). One definition of … Continue reading

## Co-Null Sets

Often in mathematics one wants to show that certain object exist. While some times the object can be explicitly constructed, this is often achieved by some indirect approach. One very common way of doing this is to apply some form … Continue reading