Notations for probability theory #

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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 measure_theory.function.conditional_expectation.

• P⟦s|m⟧ = P[s.indicator (λ 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.rn_deriv Q We note that the notation ∂P/∂Q applies to three different cases, namely, measure_theory.measure.rn_deriv, measure_theory.signed_measure.rn_deriv and measure_theory.complex_measure.rn_deriv.

• ℙ is a notation for volume on a measured space.