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Let T be a proposition. Let A be a set.

Let Y(a) be a proposition for every a:-A.

Find an equivalent condition for:

(1) T => [ /\a:-A Y(a) ].

Let Y(a) be a proposition for every a:-A.

Find an equivalent condition for:

(1) T => [ /\a:-A Y(a) ].

(2) /\a:-A [ T => Y(a) ]

Let T be a proposition. Let A be a set.

Let Y(a) be a proposition for every a:-A.

Find an equivalent condition for:

(1) /\a:-A [ T => Y(a) ]

Let Y(a) be a proposition for every a:-A.

Find an equivalent condition for:

(1) /\a:-A [ T => Y(a) ]

(2) T => [ /\a:-A Y(a) ]

Let W be a s-ring in X.

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

Prove that N(y) c W <=> ???.

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

Prove that N(y) c W <=> ???.

N(y) c W <=> W^ = W

In the proof use a theorem which works in a simpler setting.

Simply recall that theorem.

page 78 in 2nd measure

In the proof use a theorem which works in a simpler setting.

Simply recall that theorem.

page 78 in 2nd measure

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

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

(2) /\B:-W ???

Find an equivalent condition.

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

(2) /\B:-W ???

Find an equivalent condition.

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

page 78 in 2nd measure

page 78 in 2nd measure

Let W be a s-ring in X.

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

Prove that W^ = W <=> ???.

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

Prove that W^ = W <=> ???.

W^ = W <=> N(y) c W

In the proof use a theorem which works in a simpler setting.

Simply recall that theorem.

page 78 in 2nd measure

In the proof use a theorem which works in a simpler setting.

Simply recall that theorem.

page 78 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.

Let A c |R^p and y*(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.

Let A c |R^p and y*(A) = 0.

A :- MF(y)

page 79 in 2nd measure

page 79 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.

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.

For every A:-M(y) there exist two Borel sets F,G

such that F c A c G and y(G\F)=y(G\A)=y(A\F)=0.

Hence M(y) c B^, if B denotes the set of all Borel sets

and B^ as on page 72 in 2nd measure.

page 81 in 2nd measure

such that F c A c G and y(G\F)=y(G\A)=y(A\F)=0.

Hence M(y) c B^, if B denotes the set of all Borel sets

and B^ as on page 72 in 2nd measure.

page 81 in 2nd measure

Let X be a complete metric space. Let W be a ring in X.

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

Let A:-W and suppose that A is totally bounded.

How can we express y(A) in a very interesting way?

Prove the answer.

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

Let A:-W and suppose that A is totally bounded.

How can we express y(A) in a very interesting way?

Prove the answer.

y(A) = inf { +(n=1 to oo) y(E[n]) : E[n] is an open subset of W
and A c \\//E[n] }.

page 91 in 2nd measure

page 91 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 M(y) denote the collection of all y-measurable subsets of |R^p.

Let |B denote the set of all Borel subsets of |R^p.

Prove that

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

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

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

Let |B denote the set of all Borel subsets of |R^p.

Prove that

page 81 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 M(y) denote the collection of all y-measurable subsets of |R^p.

Every y-measurable set is a disjoint union of _______ and

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

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

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

Every y-measurable set is a disjoint union of _______ and

Every y-measurable set is a disjoint union of a Borel set and
of a set of measure zero.

page 82 in 2nd measure

page 82 in 2nd measure