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.

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

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

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