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/- | |
Copyright (c) 2021 RΓ©my Degenne. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: RΓ©my 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 β€ m0`, | |
- `P[X] = β« a, X a βP` | |
- `πΌ[X] = β« a, X a` | |
- `πΌ[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 =ββ Y`: `X =α΅[volume] Y` | |
- `X β€ββ Y`: `X β€α΅[volume] Y` | |
- `βP/βQ = P.rn_deriv Q` | |
We note that the notation `βP/βQ` applies to three different cases, namely, | |
`measure_theory.measure.rn_deriv`, `measure_theory.signed_measure.rn_deriv` and | |
`measure_theory.complex_measure.rn_deriv`. | |
- `β` is a notation for `volume` on a measured space. | |
-/ | |
open measure_theory | |
-- We define notations `πΌ[f|m]` for the conditional expectation of `f` with respect to `m`. | |
localized "notation `πΌ[` X `|` m `]` := | |
measure_theory.condexp m measure_theory.measure_space.volume X" in probability_theory | |
localized "notation P `[` X `]` := β« x, X x βP" in probability_theory | |
localized "notation `πΌ[` X `]` := β« a, X a" in probability_theory | |
localized "notation X `=ββ`:50 Y:50 := X =α΅[measure_theory.measure_space.volume] Y" | |
in probability_theory | |
localized "notation X `β€ββ`:50 Y:50 := X β€α΅[measure_theory.measure_space.volume] Y" | |
in probability_theory | |
localized "notation `β` P `/β`:50 Q:50 := P.rn_deriv Q" in probability_theory | |
localized "notation `β` := measure_theory.measure_space.volume" in probability_theory | |