Notations for probability theory #

This file defines the following notations, for functions X,Y, measures P, Q defined on a measurable space m0, and another measurable space structure m with hm : m ≤ m0,

P[X] = ∫ a, X a ∂P
𝔼[X] = ∫ a, X a
𝔼[X|m]: conditional expectation of X with respect to the measure volume and the measurable space m. The similar P[X|m] for a measure P is defined in MeasureTheory.Function.ConditionalExpectation.Basic.
P⟦s|m⟧ = P[s.indicator (fun ω => (1 : ℝ)) | m], conditional probability of a set.
X =ₐₛ Y: X =ᵐ[volume] Y
X ≤ₐₛ Y: X ≤ᵐ[volume] Y
∂P/∂Q = P.rnDeriv Q We note that the notation ∂P/∂Q applies to three different cases, namely, MeasureTheory.Measure.rnDeriv, MeasureTheory.SignedMeasure.rnDeriv and MeasureTheory.ComplexMeasure.rnDeriv.
ℙ is a notation for volume on a measured space.