Let us begin from a field σ(ρ) generated by one function, called ρ, on a measure space with a measure μ. We always consider σ(ρ) complete with respect to μ, that is, with all zero measure sets added. Denote the restriction of μ to σ(ρ) by μ_ρ. Denote further on the indicator of set E by 1_E.
By the very definition, the conditional expectation f_ρ of a function f with respect to σ(ρ) is the Radon-Nikodym derivative f_ρ = dfμ / dμ_ρ. How to compute it?
First note that σ(ρ) is generated by sets E_{ (a,b) } = {a 0} μ (f 1_{ E_ε(y) } ) / μ (E_ε(y) )
For y such that μ( E_ε(y) ) = 0 for some ε >0, we may define G(y) to be anything
(say, zero) since it will not change the function G(ρ).
Some argument necessary to show that G is Borel, follows from approximating ρ with step functions. I can elaborate on this if necessary.
With a field generated by several functions you proceed similarly, replacing an interval
(a,b) with a rectangular (a_1,b_1)\times (a_2,b_2).....\times(a_n,b_n).
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