1 Construction of measures 3 1.

Preface In these notes we explain the measure theoretic foundations of modern probability. The notes are used during a course that had as one of its principal aims a swift introduction to measure theory as far as it is needed in modern probability, e.g. to define concepts as conditional expectation and to prove limit theorems for martingales. Everyone with a basic notion of mathematics and prob...

1 Sigma-Algebra: Describing measurable sets 6 1.1 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Restricted algebras . . . . . . . . . . . . . . . . . . . . 10 1.1.3 Sigma Algebras . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 Binary Unions . . . . . . . . . . . . . . . . . . . . . . 12 1.1...

1 Describing measurable sets 8 1.1 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Restricted algebras . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Sigma Algebras . . . . . . . . . . . . . . . . . . . . . . 12 1.1.4 Binary Unions . . . . . . . . . . . . . . . . . . . . . . 14 1.1.5 Initial Sigm...

The conventional definition of a topological metric over a space specifies properties of any measure of “how separated” two points in that space are. Here it is shown how to extend that definition, and in particular the triangle inequality, to concern arbitrary numbers of points. Such a measure of how separated the points within a collection are can be bootstrapped, to measure “how separated” f...

A sequence of real numbers an converges to a limit a if and only if, for each 2 > 0, the sequence an eventually lies within a ball of radius 2 centered at a. It’s okay if the first few (or few million) terms lie outside that ball— and the number of terms that do lie outside the ball may depend on how big 2 is (if 2 is small enough it will take millions of terms before the remaining sequence lie...