/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import measure_theory.function.conditional_expectation

/-! # 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
  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`.

- `ℙ` is a notation for `volume` on a measured space.
-/

open measure_theory

-- We define notations `𝔼[f|m]` for the conditional expectation of `f` with respect to `m`.
localized "notation `𝔼[` X `|` m `]` :=
  measure_theory.condexp m measure_theory.measure_space.volume X" in probability_theory

localized "notation P `[` X `]` := ∫ x, X x ∂P" in probability_theory

localized "notation `𝔼[` X `]` := ∫ a, X a" in probability_theory

localized "notation X `=ₐₛ`:50 Y:50 := X =ᵐ[measure_theory.measure_space.volume] Y"
  in probability_theory

localized "notation X `≤ₐₛ`:50 Y:50 := X ≤ᵐ[measure_theory.measure_space.volume] Y"
  in probability_theory

localized "notation `∂` P `/∂`:50 Q:50 := P.rn_deriv Q" in probability_theory

localized "notation `ℙ` := measure_theory.measure_space.volume" in probability_theory