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Let W be a s-ring in X.

Let y : W -> [0,oo] be countably additive, y(O)=0.

Define a s-ring in X that contains W.

Don't prove that it is a s-ring, in this item.

(In this SM collection, it will be denoted W^.)

Let y : W -> [0,oo] be countably additive, y(O)=0.

Define a s-ring in X that contains W.

Don't prove that it is a s-ring, in this item.

(In this SM collection, it will be denoted W^.)

Let W^ = { EcX : \/A,B:-W AcEcB and y(B\A)=0 }.

page 72 in 2nd measure

page 72 in 2nd measure

Let W be a s-ring in X.

Let y : W -> [0,oo] countably additive, y(O)=0.

Prove that N(y^) c W^.

Let y : W -> [0,oo] countably additive, y(O)=0.

Prove that N(y^) c W^.

page 73 in 2nd measure

Let F be a s-algebra in X.

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

What can we conclude about the sets:

{x:-X : f(x) = oo}, {x:-X : f(x) = -oo} ?

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

What can we conclude about the sets:

{x:-X : f(x) = oo}, {x:-X : f(x) = -oo} ?

They belong to F.

page 73 in 2nd measure

page 73 in 2nd measure

Let F be a s-algebra in X.

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

What can we conclude about the set

{x:-X : -oo < f(x) < oo } ?

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

What can we conclude about the set

{x:-X : -oo < f(x) < oo } ?

It belongs to F.

page 73 in 2nd measure

page 73 in 2nd measure

Let F be a s-algebra in X.

Suppose that F is not equal to P(X).

Prove that there exists a function X -> |R*

that is not measurable.

Suppose that F is not equal to P(X).

Prove that there exists a function X -> |R*

that is not measurable.

page 74 in 2nd measure

Let F be a s-algebra in X. Let E:-F.

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

Prove that /\a:-|R {x:-E : f(x)>a} ???.

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

Prove that /\a:-|R {x:-E : f(x)>a} ???.

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

page 74 in 2nd measure

page 74 in 2nd measure

Let F be a s-algebra in X.

Let /\n:-|N f[n] : X - > |R* be measurable.

Let X be a countable disjoint union of sets in F, X = \\//(n:- |N) E[n].

Let f : X -> |R* be defined as follows: x:-E[n] ==> f(x) := f[n](x).

What can we conclude about function f?

Let /\n:-|N f[n] : X - > |R* be measurable.

Let X be a countable disjoint union of sets in F, X = \\//(n:- |N) E[n].

Let f : X -> |R* be defined as follows: x:-E[n] ==> f(x) := f[n](x).

What can we conclude about function f?

It is measurable.

Notice here that X could be written as a finite disjoint union.

page 75 in 2nd measure

page 84 in 2nd measure

Notice here that X could be written as a finite disjoint union.

page 75 in 2nd measure

page 84 in 2nd measure

Let W be a ring in X. Let y:W->|R* be a additive.

Suppose that whenever {A[n]} is a decreasing sequence of sets in W,

such that its intersection is empty, then lim(n->oo) y(A[n]) = 0.

What can we conclude from this?

Suppose that whenever {A[n]} is a decreasing sequence of sets in W,

such that its intersection is empty, then lim(n->oo) y(A[n]) = 0.

What can we conclude from this?

y is countably additive on W

page 75 in 2nd measure

page 75 in 2nd measure

Let W be a ring in X. Let y:W->|R* be countably additive.

Let A[n] be a decreasing sequence of sets in W, having empty intersection.

Suppose that y(A[1]) :- |R.

What can we conclude from this?

Let A[n] be a decreasing sequence of sets in W, having empty intersection.

Suppose that y(A[1]) :- |R.

What can we conclude from this?

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

page 77 in 2nd measure

page 77 in 2nd measure

Let W be a ring in X. Let y : W -> |R*.

(1) /\B:-W /\AcB [ y(B) = 0 ==> A:-W ]

(2) /\EcX ???

Find an equivalent condition.

(1) /\B:-W /\AcB [ y(B) = 0 ==> A:-W ]

(2) /\EcX ???

Find an equivalent condition.

(2) /\EcX [ \/A,B:-W AcEcB and y(B\A)=0 ] => E:-W

page 78 in 2nd measure

page 78 in 2nd measure