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Let E be the collection of all elementary subsets of |R^p.

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

YES.

Every open subset of |R^p is a countable union of bounded open intervals.

page 131 in 1st measure

Every open subset of |R^p is a countable union of bounded open intervals.

page 131 in 1st measure

Let E be the collection of all elementary subsets of |R^p.

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

YES.

Every open subset of |R^p is a countable union of bounded open intervals.

Hence M(y) contains all open sets.

And since M(y) is an algebra, it also contains all closed sets.

page 131 in 1st measure

Every open subset of |R^p is a countable union of bounded open intervals.

Hence M(y) contains all open sets.

And since M(y) is an algebra, it also contains all closed sets.

page 131 in 1st measure

What are Borel sets?

Let (X,G) be a topological space.

The s-algebra generated by G is called the s-algebra of Borel sets.

The s-algebra generated by G is called the s-algebra of Borel sets.

Let E be the collection of all elementary subsets of |R^p.

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

Yes.

Let G denote the collection of all open subsets of |R^p.

Then G c M(y).

M(y) is a s-algebra, hence s(G) c M(y).

s(G) is the s-algebra generated by G

s(G) is the collection of all Borel sets

Let G denote the collection of all open subsets of |R^p.

Then G c M(y).

M(y) is a s-algebra, hence s(G) c M(y).

s(G) is the s-algebra generated by G

s(G) is the collection of all Borel sets

Let F be a s-ring in X.

Let f : X -> |R*.

(1) /\a:-|R {x:-X : f(x)>a} :- F

(2) /\a:-|R {x:-X : f(x)>=a} :- F

Prove that (1) => (2).

Let f : X -> |R*.

(1) /\a:-|R {x:-X : f(x)>a} :- F

(2) /\a:-|R {x:-X : f(x)>=a} :- F

Prove that (1) => (2).

page 132 in 1st measure

Let F be a s-algebra in X.

Let f : X -> |R*.

(2) /\a:-|R {x:-X : f(x)>=a} :- F

(3) /\a:-|R {x:-X : f(x)<a} :- F

Prove that (2) => (3).

Let f : X -> |R*.

(2) /\a:-|R {x:-X : f(x)>=a} :- F

(3) /\a:-|R {x:-X : f(x)<a} :- F

Prove that (2) => (3).

page 132 in 1st measure

Let F be a s-ring in X.

Let f : X -> |R*.

(3) /\a:-|R {x:-X : f(x)<a} :- F

(4) /\a:-|R {x:-X : f(x)<=a} :- F

Prove that (3)=>(4).

Let f : X -> |R*.

(3) /\a:-|R {x:-X : f(x)<a} :- F

(4) /\a:-|R {x:-X : f(x)<=a} :- F

Prove that (3)=>(4).

page 132 in 1st measure

Let F be a s-algebra in X.

Let f : X -> |R*.

(4) /\a:-|R {x:-X : f(x)<=a} :- F

(1) /\a:-|R {x:-X : f(x)>a} :- F

Prove that (4)=>(1).

Let f : X -> |R*.

(4) /\a:-|R {x:-X : f(x)<=a} :- F

(1) /\a:-|R {x:-X : f(x)>a} :- F

Prove that (4)=>(1).

page 132 in 1st measure

Let F be a s-algebra in X.

Let f : X -> |R*.

What does it mean that "f is measurable" ?

Let f : X -> |R*.

What does it mean that "f is measurable" ?

(1) /\a:-|R {x:-X : f(x)>a} :- F

(2) /\a:-|R {x:-X : f(x)>=a} :- F

(3) /\a:-|R {x:-X : f(x)<a} :- F

(4) /\a:-|R {x:-X : f(x)<=a} :- F

these four are equivalent

page 132,133 in 1st measure

(2) /\a:-|R {x:-X : f(x)>=a} :- F

(3) /\a:-|R {x:-X : f(x)<a} :- F

(4) /\a:-|R {x:-X : f(x)<=a} :- F

these four are equivalent

page 132,133 in 1st measure

Let F be a s-algebra in X.

Let f : X -> |R* be measurable.

What can we conclude about function |f| ?

Let f : X -> |R* be measurable.

What can we conclude about function |f| ?

|f| is measurable

page 133 in 1st measure

page 133 in 1st measure