Course on Inverse Problems Albert Tarantola Lesson V: b) Advanced - - PowerPoint PPT Presentation

Princeton University

Department of Geosciences

Course on Inverse Problems

Albert Tarantola

Lesson V: b) Advanced Probability Notions (Mapping of Probabilities)

The following viewgraphs are a simplified version

• f my note

http://arxiv.org/pdf/0810.4749 . This supplementary information is not an integral part

• f the course.

Radon-Nikodym theorem

Assume given a measurable space (Ω, F) , and a measure µ

• n (Ω, F) that is σ-finite. Then, (Ω, F, µ) is called a σ-finite

measure space. Let ν be a second measure on (Ω, F) . The fol- lowing assertions are equivalent (Radon-Nikodym theorem).

• The measure ν is absolutely continuous with respect to µ .
• There is a µ-almost everywhere unique function from Ω into

[0, ∞) , denoted dν/dµ , such that

ν[F] =

• F

dν dµ dµ for every F ∈ F . The function dν/dµ is called the Radon-Nikodym density as- sociated with ν by µ . If a σ-finite measure µ is such that µ[Ω] = 1 , one says that µ is a probability measure, and the measure µ[F] of some set F ∈ F is then called the probability of the set F .

Intersection of measures

Given a σ-finite measure space (Ω, F, µ) , consider two σ-finite measures ν1 and ν2 , that are absolutely continuous with re- spect to the base measure µ . Definition: The intersection of the two measures ν1 and ν2 , is the measure denoted ν1 ∩ ν2 whose Radon-Nikodym density is d(ν1 ∩ ν2) dµ

= 1

n dν1 dµ dν2 dµ . with n =

Ω dν1 dµ dν2 dµ dµ .

intersection

X X X X

Reciprocal image of a measure

Let (X, E) and (Y, F) be two measurable spaces, and ϕ : X → Y a measurable mapping. Two measures µ and ν are introduced (to be considered as base measures) such that

(X, E, µ) and (Y, F, ν) are σ-finite measure spaces.

Definition: To every measure τ on (Y, F) that is absolutely con- tinuous with respect to ν , is associated a measure on (X, E) , called the reciprocal image of τ , denoted ϕ-1[ τ ] , and defined via d(ϕ-1[ τ ]) dµ

= 1

n dτ dν ◦ ϕ

• ,

with n =

X dτ dν ◦ ϕ dµ .

Image of a measure

Let (X, E) and (Y, F) be two measurable spaces, and ϕ : X → Y a measurable mapping. Definition: To every measure π on (X, E) , is associated a mea- sure on (Y, F) , denoted ϕ[π] , called the image measure: ϕ[π] = π ◦ ϕ-1 , i.e., explicitly, (ϕ[π])[F] = π[ ϕ-1[F] ] for every F ∈ F . Comment: To have an intuitive idea of the notion “image of a measure”, consider a collection of elements x1, x2, x3, . . . of X that are independent sample elements of the measure π . Then, it is easy to see that the elements ϕ(x1), ϕ(x2), ϕ(x3), . . .

• f Y are independent sample elements of the measure ϕ[π] .

In fact, this property alone may suggest introducing the notion

• f an image of a measure.

Compatibility property

Let (X, E, µ) and (Y, F, ν) be two σ-finite measure spaces, and ϕ : X → Y be a measurable mapping. Let π be a measure over (X, E) that is σ-finite, and τ a measure over

(Y, F) , that is absolutely continuous with respect to the base

measure ν . Theorem: One always has ϕ[ π ∩ π′ ] = τ′ ∩ τ where

• π′ = ϕ-1[ τ ]

τ′ = ϕ[π] .

X Y

X X Y Y i

• 1

i

X X Y Y X Y i

• 1

i intersection intersection

X X Y Y X Y i

• 1

i image intersection intersection

X X Y Y X Y i

• 1

i image intersection intersection