Measurable space

Sigma-algebra

Measure

Measure space

Measurable function

Measure Spaces

Theorem: Kolmogorov extension theorem to extend algebras to Sigma-algebras, giving rise to the Lebesgue measure

Integration

Integrating a function w.r.t. a measure. Given a Measure space, we can use the characteristic function of a set, to define "simple" functions defined as taking a certain value $c_i$ for $x$ that lies in any of a family of {elements $A_i$ of the sigma-algebra (which is a subset)}, which are disjoint. The integral is just the area under the graph, defined as $\int d\mu :=\sum_{i=1}^N c_i \mu(A_i)$. Can extend for integral of non-negative Measurable functions

Space of measures

Space of Borel probability measures

Would be nice to give this space some structure, like a Topology, which defines notions of closeness.

Natural topology: Weak-star topology, comes from Functional analysis.

(PP 1.1) Measure theory: Why measure theory - The Banach-Tarski Paradox

(PP 1.2) Measure theory: Sigma-algebras

Ergodic Theory - Stefano Luzzatto - Lecture 01