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BST 401 Probability Theory

Xing Qiu Ha Youn Lee

Department of Biostatistics and Computational BiologyUniversity of Rochester

October 5, 2010

Qiu, Lee BST 401

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Outline

1 Product Measures and Fubinis Theorem

2 Kolmogorovs Extension Theorem

Qiu, Lee BST 401

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Product measure space

The usual Euclidean space equipped with the usual

Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).

Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).

Qiu, Lee BST 401

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Product measure space

The usual Euclidean space equipped with the usual

Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).

Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).

Qiu, Lee BST 401

http://goforward/http://find/http://goback/

8/8/2019 Probability Theory Presentation 09

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Product measure space

The usual Euclidean space equipped with the usual

Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).

Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).

Qiu, Lee BST 401

http://goforward/http://find/http://goback/

8/8/2019 Probability Theory Presentation 09

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Product measure space

The usual Euclidean space equipped with the usual

Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).

Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).

Qiu, Lee BST 401

http://goforward/http://find/http://goback/

8/8/2019 Probability Theory Presentation 09

7/45

Product measure space

The usual Euclidean space equipped with the usual

Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).

Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).

Qiu, Lee BST 401

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The Classical Product Measure Theorem

Calculus analogy: double integral. Draw a diagram.

For E F1F2, let Ex (Ey) be the cross-section at x (y),

Ex = {y : (x, y) E} , Ey = {x : (x, y) E} .

The measure of E can be calculated by iterated integration:

(E) =

1

2(Ex)d1(x) =

2

1(Ey)d2(y).

There is one and only one measure which satisfies(A1 A2) = 1(A1)2(A2). (There is a mathematicalterminology for this property. () is called the tensorproduct of 1() and (), denoted by 1 2).

Qiu, Lee BST 401

http://goforward/http://find/http://goback/

8/8/2019 Probability Theory Presentation 09

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The Classical Product Measure Theorem

Calculus analogy: double integral. Draw a diagram.

For E F1F2, let Ex (Ey) be the cross-section at x (y),

Ex = {y : (x, y) E} , Ey = {x : (x, y) E} .

The measure of E can be calculated by iterated integration:

(E) =

1

2(Ex)d1(x) =

2

1(Ey)d2(y).

There is one and only one measure which satisfies(A1 A2) = 1(A1)2(A2). (There is a mathematicalterminology for this property. () is called the tensorproduct of 1() and (), denoted by 1 2).

Qiu, Lee BST 401

http://goforward/http://find/http://goback/

8/8/2019 Probability Theory Presentation 09

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The Classical Product Measure Theorem

Calculus analogy: double integral. Draw a diagram.

For E F1F2, let Ex (Ey) be the cross-section at x (y),

Ex = {y : (x, y) E} , Ey = {x : (x, y) E} .

The measure of E can be calculated by iterated integration:

(E) =

1

2(Ex)d1(x) =

2

1(Ey)d2(y).

There is one and only one measure which satisfies(A1 A2) = 1(A1)2(A2). (There is a mathematicalterminology for this property. () is called the tensorproduct of 1() and (), denoted by 1 2).

Qiu, Lee BST 401

http://goforward/http://find/http://goback/

8/8/2019 Probability Theory Presentation 09

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Fubinis Theorem

Motivation: f(x, y) defined on D = A B. How do we knowthat

D f(x, y)dxdy can be computed by iterated

integration:

A

B

f(x, y)dydx?

The answer: D |f(x, y)

|dxdy