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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 sring 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 sring 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 sring 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 sring 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 semiring? 
Yes. page 60 in 1st measure 
Consider the pdimensional 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 pdimensional 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 pdimensional Euclidean space R^p. In the context of measure theory, consider the collection of all intervals in R^p. Is it a semiring? 
Yes. Recall that the intervals in R form a semiring. Then recall that if X,Y are semirings, then { A x B : A:X, B:Y } is also a semiring. page 101 in 1st measure 