Talk:Algebra of random variables

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Shouldn't the *-algebra be a Von Neumann algebra to take into account infinite dimensional spaces?

Conditional expectation

Is there a similar axiomatization for conditional expectation? -- Spireguy (talk) 02:36, 26 March 2010 (UTC)[reply]

One definition of the quantum (non-commutative) version of a conditional expectation is

"Let there be a quantum probability space (N,ρ) and a commutative von Neumann sub-algebra C ⊂ N. Then the quantum conditional expectation π( · ) is a map from C′ (the commutant of C) to C such that ρ (π(X)Z) = ρ (XZ) for all X ∈ C′, Z ∈ C."

So I'd hazard a guess that an appropriate algebraic definition for the (commutative) conditional expectation would be

"If N is the algebra of all random variables considered, and C is a (von Neumann) sub-algebra (relating to the measurement outcome), then the conditional expectation is a map P from N to C such that E[P(X)Z] = E[XZ] for all X in N and Z in C"

Thoughts? -- S.Wilson (talk) 10:46, 9 May 2011 (UTC)[reply]