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Let Y be an outer measure on X.

Define the corresponding pseudometric on P(X).

Prove that it is a pseudometric.

Define the corresponding pseudometric on P(X).

Prove that it is a pseudometric.

If A,B c X, then let d(A,B) = Y(A+B).

To prove the triangle inequality, use

A + B c (A + C) u (C + B).

REMARK:

This d can assume plus infinity, hence it is controversial

To prove the triangle inequality, use

A + B c (A + C) u (C + B).

REMARK:

This d can assume plus infinity, hence it is controversial

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

d(A u B , C u D) <= ???

and let d be the corresponding pseudometric on P(X).

d(A u B , C u D) <= ???

d(A u B , C u D) <= d(A,C) + d(B,D)

page 94 in 1st measure

page 94 in 1st measure

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

d(A n B , C n D) <= ???

and let d be the corresponding pseudometric on P(X).

d(A n B , C n D) <= ???

d(A n B , C n D) <= d(A,C) + d(B,D)

page 94 in 1st measure

page 94 in 1st measure

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

d(A \ B , C \ D) <= ???

and let d be the corresponding pseudometric on P(X).

d(A \ B , C \ D) <= ???

d(A \ B , C \ D) <= d(A,C) + d(B,D)

page 94 in 1st measure

page 94 in 1st measure

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

d(A + B , C + D) <= ???

and let d be the corresponding pseudometric on P(X).

d(A + B , C + D) <= ???

d(A + B , C + D) <= d(A,C) + d(B,D)

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

In this context,

what does it mean that a sequence of sets converges to a set?

and let d be the corresponding pseudometric on P(X).

In this context,

what does it mean that a sequence of sets converges to a set?

/\n:-|N A[n] c X and A c X

A[n] -> A <=> lim d(A[n],A) = 0 <=> lim Y(A[n] + A) = 0

A[n] -> A <=> lim d(A[n],A) = 0 <=> lim Y(A[n] + A) = 0

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] u B[n] -> A u B.

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] u B[n] -> A u B.

use:

d(A[n] u B[n] , A u B) <= d(A[n],A) + d(B[n],B)

page 95 in 1st measure

d(A[n] u B[n] , A u B) <= d(A[n],A) + d(B[n],B)

page 95 in 1st measure

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] n B[n] -> A n B.

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] n B[n] -> A n B.

use:

d(A[n] n B[n] , A n B) <= d(A[n],A) + d(B[n],B)

d(A[n] n B[n] , A n B) <= d(A[n],A) + d(B[n],B)

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] \ B[n] -> A \ B.

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] \ B[n] -> A \ B.

use:

d(A[n] \ B[n] , A \ B) <= d(A[n],A) + d(B[n],B)

d(A[n] \ B[n] , A \ B) <= d(A[n],A) + d(B[n],B)

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] + B[n] -> A + B.

and let d be the corresponding pseudometric on P(X).

Let A[n] -> A and B[n] -> B.

Prove that A[n] + B[n] -> A + B.

use:

d(A[n] + B[n] , A + B) <= d(A[n],A) + d(B[n],B)

d(A[n] + B[n] , A + B) <= d(A[n],A) + d(B[n],B)