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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.

Prove that y* is additive on MF(y).

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.

Prove that y* is additive on MF(y).

page 119 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.

Recall that MF(y) is a ring.

Use the fact that every countable union of sets in a ring can be written as a disjoint countable union of sets in that ring.

page 120 in 1st measure

Use the fact that every countable union of sets in a ring can be written as a disjoint countable union of sets in that ring.

page 120 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.

y*(A) = +(n=1 to n=oo) y*(A[n])

Recall that MF(y) is a ring and y* is additive on MF(y)

and use this in the proof.

page 121 in 1st measure

Recall that MF(y) is a ring and y* is additive on MF(y)

and use this in the proof.

page 121 in 1st measure

Let W be a s-ring in X.

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

Let A[n] be a sequence of sets from W.

y( lim_inf A[n] ) < lim_inf y( A[n] ) < lim_sup y( A[n] ) < y( lim_sup A[n] )

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

Let A[n] be a sequence of sets from W.

y( lim_inf A[n] ) < lim_inf y( A[n] ) < lim_sup y( A[n] ) < y( lim_sup A[n] )

Let X = {1,2,3}.

Let y be the counting measure on X.

Let A[2n] = {1}, A[2n+1] = {2,3}.

We have 0 < 1 < 2 < 3.

see pages 159,160 in 1st measure

Let y be the counting measure on X.

Let A[2n] = {1}, A[2n+1] = {2,3}.

We have 0 < 1 < 2 < 3.

see pages 159,160 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.

y*(A) < oo

PROOF:

Since A:-MF(y), there is B:-E such that y*(A+B)<7, and y*(B)<oo.

Notice that A c (A+B) u B. Hence y*(A) <= y*(A+B) + y*(B) < oo.

page 122 in 1st measure

PROOF:

Since A:-MF(y), there is B:-E such that y*(A+B)<7, and y*(B)<oo.

Notice that A c (A+B) u B. Hence y*(A) <= y*(A+B) + y*(B) < oo.

page 122 in 1st measure

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.

A :- MF(y)

For the proof, use the fact below.

If A is a countable disjoint union of sets in MF(y), say A = U(n:-|N)_A[n],

then y*(A) = +(n:-|N)_y*(A[n]).

For the proof, use the fact below.

If A is a countable disjoint union of sets in MF(y), say A = U(n:-|N)_A[n],

then y*(A) = +(n:-|N)_y*(A[n]).

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.

In order to prove this, we use:

(1) MF(y) is a ring

(2) A:-M(y) ==> [ y*(A)<oo <=> A:-MF(y) ]

page 124 in 1st measure

In order to prove this, we use:

(1) MF(y) is a ring

(2) A:-M(y) ==> [ y*(A)<oo <=> A:-MF(y) ]

page 124 in 1st measure

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.

Use the fact that y* is countably additive on MF(y).

page 125 in 1st measure

Take any A[n],A :- M(y) such that \\*//(n:-|N)_A[n] = A.

If \/k:-|N y(A[k])=oo, then y(A) = oo = +(n=1 to oo) y(A[n]).

If /\k:-|N y(A[k])<oo, then /\k:-|N A[k] :- MF(y)

and use Theorem 86 on page 121 in 1st measure.

page 125 in 1st measure

Take any A[n],A :- M(y) such that \\*//(n:-|N)_A[n] = A.

If \/k:-|N y(A[k])=oo, then y(A) = oo = +(n=1 to oo) y(A[n]).

If /\k:-|N y(A[k])<oo, then /\k:-|N A[k] :- MF(y)

and use Theorem 86 on page 121 in 1st measure.

( U(t:-T) G[t] ) \ ( U(t:-T) E[t] ) c ???

( U(t:-T) G[t] ) \ ( U(t:-T) E[t] ) c U(t:-T) G[t]\E[t]

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.

page 127 in 1st measure

page 79 in 2nd measure

page 79 in 2nd measure