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Let W c P(X), A c X.

Let A^ be the set of all points which are W-adherent to A.

Let Clo(A) = //\\ {FcX: F is W-closed and A c F}.

What can we say about these two sets?

Let A^ be the set of all points which are W-adherent to A.

Let Clo(A) = //\\ {FcX: F is W-closed and A c F}.

What can we say about these two sets?

A^ = Clo(A)

page 65 in gen top

page 65 in gen top

Let W c P(X), A c X, x :- X.

What does it mean that x is sequentially W-adherent to A ?

What does it mean that x is sequentially W-adherent to A ?

There exists a sequence in A, which W-converges to x.

Let W c P(X), A c X.

What does it mean that A is sequentially W-closed ?

What does it mean that A is sequentially W-closed ?

(1) Every point, that is sequentially W-adherent to A, belongs
to A.

(2) If a sequence contained in A W-converges to x, then x belongs to A.

These two are equivalent.

page 66 in gen top

(2) If a sequence contained in A W-converges to x, then x belongs to A.

These two are equivalent.

page 66 in gen top

Let X be a set, and W c P(X), and A c X.

Let As denote the set of all points which are sequentially W- adherent to A.

Does As have to be sequentially W-closed?

Let As denote the set of all points which are sequentially W- adherent to A.

Does As have to be sequentially W-closed?

NO.

page 67 in gen top

page 67 in gen top

Let W be a ring. Let y : W -> |R* be additive. Let A,B :- W.

Suppose that B c A and y(A):-|R.

Is it possible that y(B) = oo or that y(B) = -oo ?

Suppose that B c A and y(A):-|R.

Is it possible that y(B) = oo or that y(B) = -oo ?

NO.

1) Since B c A, we have A = B u (A\B) disjointly.

2) Since W is a ring and y is additive on W, we have y(A) = y(B) + y(A\B).

3) Since y(A):-|R, both y(B) and y(A\B) must be real.

4) Hence y(B) :- |R.

page 75 in 1st measure

1) Since B c A, we have A = B u (A\B) disjointly.

2) Since W is a ring and y is additive on W, we have y(A) = y(B) + y(A\B).

3) Since y(A):-|R, both y(B) and y(A\B) must be real.

4) Hence y(B) :- |R.

page 75 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.

Define the collection MF(y) of all finitely 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.

Define the collection MF(y) of all finitely y-measurable subsets of |R^p.

MF(y) = { A c |R^p : \/{A[n]}cE lim(n->oo) y(A[n]+A) = 0 }

page 117 in 1st measure

page 117 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.

Prove that it is a ring.

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 it is a ring.

You can use the theorem from the next item.

page 118 in 1st measure

page 64 in 2nd measure

page 118 in 1st measure

page 64 in 2nd measure

Let X be a set, and let Y : P(X) -> [0,oo] satisfy:

(1) Y(O) = 0.

(2) A c B c X ==> Y(A) <= Y(B).

(3) Y(A u B) <= Y(A) + Y(B).

Let M be a ring in X.

Let F = { A c X : \/{A[n]}cM lim(n->oo) Y(A[n] + A) = 0 }.

Can we conclude that F is a ring?

(1) Y(O) = 0.

(2) A c B c X ==> Y(A) <= Y(B).

(3) Y(A u B) <= Y(A) + Y(B).

Let M be a ring in X.

Let F = { A c X : \/{A[n]}cM lim(n->oo) Y(A[n] + A) = 0 }.

Can we conclude that F is a ring?

F is a ring

page 64 in 2nd measure

page 118 in 1st measure

page 64 in 2nd measure

page 118 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.

/\B,A:-MF(y) y*(A u B) + y*(A n B) = ???

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.

/\B,A:-MF(y) y*(A u B) + y*(A n B) = ???

/\B,A:-MF(y) y*(A u B) + y*(A n B) = y*(A) + y*(B)

page 118 in 1st measure

page 118 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.

Define the collection M(y) 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.

Define the collection M(y) of all y-measurable subsets of |R^p.

M(y) = the collection of countable unions of sets from MF(y)

page 117 in 1st measure

page 117 in 1st measure