Measure Theoretic Probability
What? Why?
Probability started with simple discrete experiments that were easy to formalise without much machinery, but as the field grew and there was a need for defining probability over infinite sample spaces, (like points in a rectangle)
Our intuition for probability extends naturally of Borel Sigma algebras where unions are complementation keep you in the set while having nice properties. Measure theory offers a very useful toolset for such situations.
In analysis, measure theory can be used to extend the concept of length into area, and volume and even more abstract properties that resemble “size”
Notes
- Size of subsets of Reals
- Semifield
- Field (Measure Theory)
- Sigma Field
- Borel Sigma Field
- Measure Space
- Special Types of Measures
- Completion of a Measure Space
- Measurable Functions
- Integration (Measure Theory)
- Product Measures
- Probability
- Joint Distribution Measure and Product of Probability
- Borel’s Law of Large Numbers
- Kolmogorov’s One Series Theorem
- Kolmogorov’s Law of Large Numbers
- Conditional Expectation