Math ASCII Notation Demo

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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).
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.
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
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.
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
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 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 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.
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
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.
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]).
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.
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
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.
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.
( 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 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.
page 127 in 1st measure
page 79 in 2nd measure