At a purely formal level, one could call probability theory the study of measure spaces `Ω` with total measure one `∑Ω=1`, but that would be like calling number theory the study of strings of digits which terminate. At a practical level, the opposite is true…

it is the events and their probabilities that are viewed as being fundamental, with the sample space `Ω` being [forgotten] as much as possible, and with the random variables and expectations being viewed as derived concepts. …

However, it is possible to … abstract… one step further, and view the algebra of random variables and their expectations as being … foundational …, and ignoring both the presence of the original sample space, the algebra of events, or the probability measure.

Terry Tao 0, 5

Hat tip to Qiaochu Yuan, who echoes:

The traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space `P` and constructs random variables as certain functions `P→ℝ`. But start doing any probability and it becomes clear that the space P is de-emphasized as much as possible; the real focus of probability theory is on the algebra of random variables. It would be nice to have an approach to probability theory that reflects this.

Qiaochu is interested in extending to the noncommutative case to

As I’ve remarked elsewhere, noncommutative is normal. In life as well as in QM. In your typical Euclidean xy plane the directions don’t matter—and from this same-etry we get

• Galileo’s decomposition of motion
• tensor algebras
• a concise expression of general relativity theory as a metric
• coordinate-free representations of all these things

Practically speaking, in life, though, the direction you’re goingdoesmatter, either because the evolutionary field only has manna in the Canaanic direction, or because the playing field itself is tilted, by gravity (general relativity) or entropy or financial regulations or competition or something else.

• It’s easier to lie down than jump.
• It’s easier to fail than succeed.
• Portfolio variation in the upward direction has a notably different effect on LPs’ attitudes during phone calls than does portfolio variation in the downward direction.
• You can’t go back in time and say the words you should have said.
• Speaking of words, word order matters.

Noncommutative is normal. Looking forward to stealing these mathematicians’ great ideas on free probability!

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