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

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

Suppose that M is a s-ideal.

Does W have to be a s-ideal, too?

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

Suppose that M is a s-ideal.

Does W have to be a s-ideal, too?

YES

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 monotone.

What can we conclude about W ?

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

Suppose that M is monotone.

What can we conclude about W ?

W is also monotone

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 topology.

Does W also have to be a topology?

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

Suppose that M is a topology.

Does W also have to be a topology?

YES

page 89 in 2nd measure

page 89 in 2nd measure

Let F be a s-algebra in X.

Let f : X -> |R\{-oo,oo} be measurable.

Let g:|R->|R be such that /\a:-|R {x:-|R : g(x)>a} is Borel.

What can we conclude?

Let f : X -> |R\{-oo,oo} be measurable.

Let g:|R->|R be such that /\a:-|R {x:-|R : g(x)>a} is Borel.

What can we conclude?

gof is measurable

/\a:-|R {x:-X : g(f(x))>a} :- F

page 90 in 2nd measure

/\a:-|R {x:-X : g(f(x))>a} :- F

page 90 in 2nd measure

Every bounded sequence of rational numbers contains a Cauchy
subsequence.

How can that be proven without using the Dedekind axiom?

How can that be proven without using the Dedekind axiom?

Without using the Dedekind axiom, we can prove that every
bounded subset of |Q is totally bounded. And every sequence
contained in a totally bounded set has a Cauchy subsequence.

page 141 in golden gate

page 141 in golden gate

Let f : |R^2 -> |R have both partial derivatives: f1,f2.

(1) /\a,b:-|R |f1(a,b)| <= 2*|b-a|

(2) /\a,b:-|R |f2(a,b)| <= 2*|b-a|

(3) f(0,0) = 0

Prove that |f(5,4)| <= 1.

(1) /\a,b:-|R |f1(a,b)| <= 2*|b-a|

(2) /\a,b:-|R |f2(a,b)| <= 2*|b-a|

(3) f(0,0) = 0

Prove that |f(5,4)| <= 1.

page 77 in OLDTIMER

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.

Does MF(y) have to be a s-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.

Does MF(y) have to be a s-ring?

NO.

We know that if A:-MF(y), then y(A)<oo.

A countable union of elementary sets can be unbounded and have infinite measure.

We know that if A:-MF(y), then y(A)<oo.

A countable union of elementary sets can be unbounded and have infinite measure.

Let M be a s-algebra in X. Let f : X -> |R* be constant.

Does f have to be M-measurable?

Does f have to be M-measurable?

[ \/K:-|R* /\x:-X f(x)=K ] ==> f is measurable

page 136 in 1st measure

page 136 in 1st measure

Let M be a s-algebra in X.

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

What can we conclude about function g:X->|R* defined by g(x)=b*f(x) ?

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

What can we conclude about function g:X->|R* defined by g(x)=b*f(x) ?

Function g:X->|R* defined by g(x)=b*f(x) is M-measurable.

page 136 in 1st measure

page 136 in 1st measure

Let A[n] c X for all natural n.

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

Prove that for all x:-X

/\x:-X lim_sup 1(A[n])(x) = ???

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

Prove that for all x:-X

/\x:-X lim_sup 1(A[n])(x) = ???

/\x:-X lim_sup 1(A[n])(x) = 1(lim_sup A[n])(x)

page 11 in 1st measure

page 11 in 1st measure