Probability Measure

Delve into a comprehensive 7-hour tutorial on probability measure, covering essential topics in set theory, fields, sigma fields, measurable spaces, and probability measures. Learn about limit supremum and infimum, construct fields and sigma fields, explore set functions and their properties, and understand probability measures and their extensions. Examine concepts like outer measure, complete measure, and the Monotone Class Theorem. Study conditional probability, independence, random variables, and cumulative distribution functions. Gain insights into the Borel-Cantelli Lemmas, Erdos-Renyi Lemma, and Riemann Stieltjes Integration. Explore real-world applications through examples, including the independence of polar coordinates and the construction of non-measurable sets.

Probability Measure: 1. Set Theory.
Limit Supremum and Limit Infimum of a Sequence of Real Numbers.
Limit Supremum and Limit Infimum of Sets (part 1 of 2).
Limit Supremum and Limit Infimum of Sets (part 2 of 2).
2 Examples with limsup and liminf.
Probability Measure: 2. Fields.
How to Construct the Smallest Field Containing Sets A1,..., An.
Probability Measure: 3. Sigma Fields.
Probability Measure: 4. Measurable Spaces.
Set Functions on Measurable Spaces.
Properties of Set Functions.
Continuity of a Set Function.
A subset (Vitali set) of the Reals that is not Lebesgue measurable.
Probability Measure: 5. Probability Measure.
Extension of a probability measure from a field to a slightly larger class of sets..
Extension of a probability measure to all subsets of omega.
Outer Measure.
A probability measure on a field, F, can be extended to a probability measure on sigma(F).
Complete Measure.
Example of a completion of a measure space.
Monotone Class Theorem.
Caratheodory Extension Theorem.
1st and 2nd Borel Cantelli Lemmas.
Erdos-Renyi Lemma: Extension of the 2nd Borel-Cantelli Lemma.
Approximation Theorem (Measure Theory).
Probability Measure: 6. Conditional Probability.
Theorem of Total Probability.
Probability Measure: 7. Independence.
Show that R & Theta are Independent in Polar Coordinates.
Probability Measure: 8. Random Variable.
Probability Measure: 9. Functions of Random Variables / Vectors.
Probability Measure: 10 Cumulative Distribution Function.
Riemann Stieltjes Integration for Statisticians.
Example where both the Approximation theorem and Caratheodory Extension Theorem Fail.