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Let X be an arbitrary set. Does there exist an extended real-
valued function on P(X) that is countably additive?

Yes. The counting measure.

F : P(X) -> [0,oo]

If A c X is finite, then F(A) = the number of elements of A.

If A c X is infinite, then F(A) = oo.

Check that it is countably additive!

F : P(X) -> [0,oo]

If A c X is finite, then F(A) = the number of elements of A.

If A c X is infinite, then F(A) = oo.

Check that it is countably additive!

Let L be a countable set.

Let W = { A c L : L\A is finite }.

Describe M(W) - the monotone class generated by W.

Let W = { A c L : L\A is finite }.

Describe M(W) - the monotone class generated by W.

M(W) = P(L).

page 48 in 1st measure

page 48 in 1st measure

Let A,B be subsets of a metric space.

Suppose that A n Clo(B) is non-empty and A n B is empty.

What can we conclude from this?

Suppose that A n Clo(B) is non-empty and A n B is empty.

What can we conclude from this?

A is not open

X set, let M be a countable subset of X.

W = {E c X : M\E is finite}

Describe S(W) - the s-ring generated by W.

W = {E c X : M\E is finite}

Describe S(W) - the s-ring generated by W.

S(W) = { E c X : MnE is countable or M\E is countable }

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

Define "A is W-open" in the language of generalized convergence.

Define "A is W-open" in the language of generalized convergence.

Let Fr(A) denote the boundary of A. (see the prev item)

A is W-open <=> Fr(A) c X\A

page 47 in gen top

A is W-open <=> Fr(A) c X\A

page 47 in gen top

Express differently:

A \ //\\t:T [ B[t] ] = ???

A \ //\\t:T [ B[t] ] = ???

A \ //\\t:T [ B[t] ] = \\//t:-T [ A\B[t] ]

Express differently:

\\//t:-T [ A\B[t] ] = ???

\\//t:-T [ A\B[t] ] = ???

\\//t:-T [ A\B[t] ] = A \ //\\t:T [ B[t] ]

Express differently:

( \\//t:-T B[t] ) \ A = ???

( \\//t:-T B[t] ) \ A = ???

( \\//t:-T B[t] ) \ A = \\//t:-T ( B[t]\A )

Express differently:

\\//t:-T B[t]\A = ???

\\//t:-T B[t]\A = ???

\\//t:-T ( B[t]\A ) = ( \\//t:-T B[t] ) \ A

Express differently:

( //\\t:-T B[t] ) \ A = ???

( //\\t:-T B[t] ) \ A = ???

( //\\t:-T B[t] ) \ A = //\\t:-T ( B[t]\A )