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Let F be a s-algebra in X.

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

Prove:

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

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

Prove:

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

page 134 in 1st measure

Let F be a s-algebra in X.

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = sup(n:-|N) f[n](x).

What can we conclude about function g?

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = sup(n:-|N) f[n](x).

What can we conclude about function g?

g is measurable

page 134 in 1st measure

page 134 in 1st measure

Let F be a s-algebra in X.

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = inf(n:-|N) f[n](x).

What can we conclude about function g?

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = inf(n:-|N) f[n](x).

What can we conclude about function g?

g is measurable

page 134 in 1st measure

page 134 in 1st measure

Let F be a s-algebra in X.

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = lim_inf(n->oo) f[n](x).

What can we conclude about function g?

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = lim_inf(n->oo) f[n](x).

What can we conclude about function g?

g is measurable

page 135 in 1st measure

page 135 in 1st measure

Let F be a s-algebra in X.

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = lim_sup(n->oo) f[n](x).

What can we conclude about function g?

Let f[n] be a sequence of measurable functions X -> |R*.

Let g : X->|R* be defined by g(x) = lim_sup(n->oo) f[n](x).

What can we conclude about function g?

g is measurable

page 135 in 1st measure

page 135 in 1st measure

Decide if it's true:

/\a,b:-|R \/x,y:-|R [ x+y = a and x-y = b ]

/\a,b:-|R \/x,y:-|R [ x+y = a and x-y = b ]

true

x = (a+b)/2

y = (a-b)/2

x = (a+b)/2

y = (a-b)/2

f : |C -> |C

(1) f is diffable

(2) \/A:-|C /\z:-|C f'(z) = A*f(z)

(3) f(0) = 1

What can we conclude?

(1) f is diffable

(2) \/A:-|C /\z:-|C f'(z) = A*f(z)

(3) f(0) = 1

What can we conclude?

(4) /\a,b:-|C f(a+b) = f(a)*f(b)

hint: fix w and define g(z)=f(z)*f(w-z)

page 78 in OLDTIMER

hint: fix w and define g(z)=f(z)*f(w-z)

page 78 in OLDTIMER

Let A,B,C,D,E,F be non-empty sets.

Decide if it's true:

(A x C) u (B x D) = E x F ==> A=B or C=D

Decide if it's true:

(A x C) u (B x D) = E x F ==> A=B or C=D

NO.

A={1}; B={1,2}; C={7}; D={7,8}; E={1,2}; F={7,8}

(A x C) u (B x D) = E x F and A!=B and C!=D

A={1}; B={1,2}; C={7}; D={7,8}; E={1,2}; F={7,8}

(A x C) u (B x D) = E x F and A!=B and C!=D

Let x > 0.

(x+2) / (x+1)^2 < ???

(x+2) / (x+1)^2 < ???

(x+2) / (x+1)^2 < 1/x

Let F be a s-algebra in X.

Let f[n] : X -> |R* be a sequence of measurable functions.

What can we say about this set:

{x:-X : \/A:-|R lim(n->oo) f[n](x) = A}

Let f[n] : X -> |R* be a sequence of measurable functions.

What can we say about this set:

{x:-X : \/A:-|R lim(n->oo) f[n](x) = A}

it belongs to F

page 83 in 2nd measure

page 83 in 2nd measure