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Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
Let B,A c X and Y(A) < oo , Y(B) < oo.
|Y(A) - Y(B)| <= ???
|Y(A) - Y(B)| <= d(A,B)
|Y(A) - Y(B)| <= Y(A+B)
way (1): use d(A,O) <= d(A,B) + d(B,O).
way (2): use Y(A) <= Y(AuB) <= Y(A+B) + Y(AnB) <= Y(A+B) + Y(B)
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
Suppose that /\n:-|N Y(A[n]) < oo , Y(A) < oo.
Suppose that A[n] -> A, ( lim(n->oo) Y(A[n]+A) = 0 ).
What can we conclude?
lim Y(A[n]) = Y(A)
hint: use |Y(A[n]) - Y(A)| <= d(A[n],A)
page 96 in 1st measure
Let a : |N x |N -> [0,oo].
Consider this line:
+(n=1 to n=oo) +(k=1 to k=oo) a(n,k) = +(k=1 to k=oo) +(n=1 to n=oo) a(n,k)
Prove this line supposing that infinity does not occur in
page 59 in 2nd measure
page 97 in 1st measure
Let X[n] be a sequence of sets. Let X be the union of all X[n].
Let S[n] be a s-ring in X[n], for every n:-|N.
Let S = { E c X : /\n:-|N ( E n X[n] :- S[n] ) }.
What can we conclude about S?
S is a s-ring
page 99 in 1st measure
Let X[n] be a sequence of sets. Let X be the union of all X[n].
Let S[n] be a s-ring in X[n], for every n:-|N.
Let y[n] : S[n] -> [0,oo] be countably additive, for every n:-|N.
Let S = { E c X : /\n:-|N ( E n X[n] :- S[n] ) }.
Let Y : S -> [0,oo] be defined by Y(E) = +(n=1 to n=oo) y[n](E n X[n]).
ANSWER: Y is countably additive.
page 99 in 1st measure
By the way, S is a s-ring in X.
But that is the subject matter of the previous item.
Consider the set of all bounded intervals on the real line, including singletons and the empty set. Is it a ring?
NO.
[1,7] \ [2,6] does not belong
Consider the set of all bounded intervals on the real line, including singletons and the empty set. Is it a semi-ring?
Yes.
page 60 in 1st measure
Consider the p-dimensional Euclidean space |R^p.
In the context of measure theory,
how do we define the term "interval" in |R^p ?
An interval in |R is a bounded connected subset of |R, including the empty set.
If {P[n]}(n=1 to n=p) is a collection of intervals in |R,
then the Cartesian product P[1] x P[2] x ... x P[p] is an interval in |R^p.
Consider the p-dimensional Euclidean space |R^p.
In the context of measure theory,
consider the collection of all intervals in |R^p.
Is it a ring?
NO.
[1,7] \ [2,4] does not belong
Consider the p-dimensional Euclidean space |R^p.
In the context of measure theory,
consider the collection of all intervals in |R^p.
Is it a semi-ring?
Yes.
Recall that the intervals in |R form a semi-ring.
Then recall that if X,Y are semi-rings,
then { A x B : A:-X, B:-Y } is also a semi-ring.
page 101 in 1st measure

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