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Let f : X -> Y. Let A,B c Y.

Suppose that A c f(X) and f-1(A) c f-1(B).

Does this imply that A c B ?

Suppose that A c f(X) and f-1(A) c f-1(B).

Does this imply that A c B ?

YES

page 87 in 2nd measure

page 87 in 2nd measure

Let f : X -> Y.

Prove that

/\BcY f-1(B) = f-1(B n ???)

Prove that

/\BcY f-1(B) = f-1(B n ???)

/\BcY f-1(B) = f-1(B n f(X))

page 97 in 2nd measure

page 97 in 2nd measure

Let f : X -> Y. Let W c P(Y).

Let K = { f-1(B) : B:-W }.

Suppose that W is monotone.

Does K have to be monotone, too?

Let K = { f-1(B) : B:-W }.

Suppose that W is monotone.

Does K have to be monotone, too?

NO

page 88 in 2nd measure

f : {1,2,3,4,...} -> {0,1,-1,2,-2,3,-3,4,-4,...}; f(n)=n

W = { {1,-1}, {1,2,-2}, {1,2,3,-3}, {1,2,3,4,-4}, ... }

K = { f-1(B) : B:-W } = { {1}, {1,2}, {1,2,3}, {1,2,3,4}, ... }

W is monotone but K is not monotone

page 88 in 2nd measure

f : {1,2,3,4,...} -> {0,1,-1,2,-2,3,-3,4,-4,...}; f(n)=n

W = { {1,-1}, {1,2,-2}, {1,2,3,-3}, {1,2,3,4,-4}, ... }

K = { f-1(B) : B:-W } = { {1}, {1,2}, {1,2,3}, {1,2,3,4}, ... }

W is monotone but K is not monotone

Let F a s-algebra in X.

Let f : X -> |R* be measurable.

Prove that

/\Gc|R [ G is open => ??? ]

Let f : X -> |R* be measurable.

Prove that

/\Gc|R [ G is open => ??? ]

/\Gc|R [ G is open => f-1(G) :- F ]

page 89 in 2nd measure

page 89 in 2nd measure

Let F be a s-algebra in X.

Let f : X -> |R* be measurable.

Prove that

/\Gc|R [ ??? => f-1(G) :- F ]

Let f : X -> |R* be measurable.

Prove that

/\Gc|R [ ??? => f-1(G) :- F ]

(1) /\Gc|R [ G is open => f-1(G) :- F ]

(2) /\Gc|R [ G is Borel => f-1(G) :- F ]

page 89,90 in 2nd measure

In order to prove (2) we use (1).

To prove (1), we recall that every open set

(2) /\Gc|R [ G is Borel => f-1(G) :- F ]

page 89,90 in 2nd measure

In order to prove (2) we use (1).

To prove (1), we recall that every open set

Let f : X -> Y. Let M c P(X).

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is a ring.

What can we conclude about W ?

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is a ring.

What can we conclude about W ?

W is also a ring

page 89 in 2nd measure

page 89 in 2nd measure

Let f : X -> Y. Let M c P(X).

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is a s-ring.

What can we conclude about W ?

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is a s-ring.

What can we conclude about W ?

W is also a s-ring

page 89 in 2nd measure

page 89 in 2nd measure

Let f : X -> Y. Let M c P(X).

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is an algebra in X.

What can we conclude about W ?

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is an algebra in X.

What can we conclude about W ?

W is an algebra in Y

page 89 in 2nd measure

page 89 in 2nd measure

Let f : X -> Y. Let M c P(X).

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is a s-algebra.

What can we conclude about W ?

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is a s-algebra.

What can we conclude about W ?

W is a s-algebra in Y

page 89 in 2nd measure

page 89 in 2nd measure

Let f : X -> Y. Let M c P(X).

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is an ideal.

Does W have to be an ideal, too?

Let W = { A c Y : f-1(A) :- M }.

Suppose that M is an ideal.

Does W have to be an ideal, too?

YES.

page 89 in 2nd measure

page 89 in 2nd measure