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What does it mean that (X,F,y) is a measure space in MRW math
materials?

(1) X is a non-empty set

(2) F is a s-algebra in X

(3) y : F -> [0,oo]

(4) y is countably additive and y(O)=0

page 102 in 2nd measure

(2) F is a s-algebra in X

(3) y : F -> [0,oo]

(4) y is countably additive and y(O)=0

page 102 in 2nd measure

If f : X -> |R*, what does (f+) mean?

(f+) : X -> [0,oo]

/\x:-X (f+)(x) = max( 0, f(x) )

page 168 in 1st measure

page 102 in 2nd measure

/\x:-X (f+)(x) = max( 0, f(x) )

page 168 in 1st measure

page 102 in 2nd measure

If f : X -> |R*, what does (f-) mean?

(f-) : X -> [0,oo]

/\x:-X (f-)(x) = max( 0, -f(x) )

page 168 in 1st measure

page 102 in 2nd measure

/\x:-X (f-)(x) = max( 0, -f(x) )

page 168 in 1st measure

page 102 in 2nd measure

Let a,b :- |R*.

Suppose that (a+) = (b+).

What can we conclude?

Suppose that (a+) = (b+).

What can we conclude?

( a<=0 and b<=0 ) or a=b

page 104 in 2nd measure

page 104 in 2nd measure

Let a,b :- |R*.

Suppose that (a-) = (b-).

What can we conclude?

Suppose that (a-) = (b-).

What can we conclude?

(a>=0 and b>=0) or a=b

page 104 in 2nd measure

page 104 in 2nd measure

Let a,b :- |R*.

Suppose that (a+)=(b+) and (a-)=(b-).

Is this enough to conclude that a=b ?

Suppose that (a+)=(b+) and (a-)=(b-).

Is this enough to conclude that a=b ?

YES

page 104 in 2nd measure

page 104 in 2nd measure

Let a,b :- |R*.

Suppose that (a+)!=(b+) or (a-)!=(b-).

What can we conclude from this?

Suppose that (a+)!=(b+) or (a-)!=(b-).

What can we conclude from this?

a != b

page 104 in 2nd measure

page 104 in 2nd measure

Let (X,F,y) be a measure space.

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

Suppose that (f+)~(g+) and (f-)~(g-).

Is this enough to conclude that f~g ?

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

Suppose that (f+)~(g+) and (f-)~(g-).

Is this enough to conclude that f~g ?

YES

page 105 in 2nd measure

page 105 in 2nd 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.

(1) Ac|R^p ; /\(n:-|N) A[n]:-MF(y)

(2) lim(n->oo) y*(A[n]+A)=0

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.

(1) Ac|R^p ; /\(n:-|N) A[n]:-MF(y)

(2) lim(n->oo) y*(A[n]+A)=0

A:-MF(y)

page 21 in 2nd measure

page 21 in 2nd measure

(1) f : X -> |R*.

(2) n:-|N

(3) a[1], ..., a[n] :- |R

(4) A[1], ..., A[n] c X , some of them may be empty

(5) X = \\*//(k=1 to n) A[k]

(6) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])(x)

(2) n:-|N

(3) a[1], ..., a[n] :- |R

(4) A[1], ..., A[n] c X , some of them may be empty

(5) X = \\*//(k=1 to n) A[k]

(6) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])(x)

f(X) c { a[1] , ... , a[n] } c |R

f is a simple function

page 248 in 2nd measure

f is a simple function

page 248 in 2nd measure