# mathlibdocumentation

probability_theory.notation

# 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|hm]: conditional expectation of X with respect to the measure volume and the measurable space m. The similar P[X|hm] for a measure P is defined in measure_theory.function.conditional_expectation.
• 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.

TODO: define the notation ℙ s for the probability of a set s, and decide whether it should be a value in ℝ, ℝ≥0 or ℝ≥0∞.