Probability Theory

Measure Theory

I include some posts on basic measure theory and sigma-algebras. This is mainly for reference, as sometimes it can be difficult to find good online references which state the definitions and prove the fundamental results in the necessary generality. The utility of the basic theory is that it provides techniques and results which can be applied in more advanced situations.

• Essential Suprema
• The Monotone Class Theorem
• The Functional Monotone Class Theorem

Quantum Probability

I include some posts on quantum theory. While this subject is generally classed as a part of physics, it is also an application of noncommutative probability theory.
See, also, the posts on algebraic probability, which develops some of the underlying maths in a more rigorous way.

• Quantum Entanglement
• Quantum Entanglement States
• Quantum Coin Tossing

Algebraic Probability

Here, I look at algebraic approaches to probability theory, which are in contrast to the classical Kolmogorov axiomatization in terms of sigma-algebras and probability measures. This includes extensions to the noncommutative probability spaces used in quantum theory.

• Algebraic Probability
• Algebraic Probability (continued)
• Algebraic Probability: Quantum Theory
• *-Algebras
• States on *-Algebras
• Homomorphisms of *-Probability Spaces
• The GNS Representation
• Operator Topologies
• Normal Maps
• Noncommutative Probability Spaces
• Completions of *-Probability Spaces

Other Probability Posts

Probability related posts which do not fit into the categories above are listed here.

• Manipulating the Normal Distribution
• Multivariate Normal Distributions
• Independence of Normals
• The Riemann Zeta Function and Probability Distributions
• Rademacher Series
• The Khintchine Inequality
• The Gaussian Correlation Conjecture
• The Gaussian Correlation Conjecture 2
• The Gaussian Correlation Inequality