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(1) W is a s-algebra in X

(2) f,g : X -> |R \ {-oo,oo} are W-measurable

(3) F : |R^2 -> |R is continuous

(4) h : X -> |R , h(x) = F(f(x),g(x))

What can we conclude from this?

(2) f,g : X -> |R \ {-oo,oo} are W-measurable

(3) F : |R^2 -> |R is continuous

(4) h : X -> |R , h(x) = F(f(x),g(x))

What can we conclude from this?

h is W-measurable

page 137 in 1st measure

page 137 in 1st measure

(1) D is an open connected subset of |C

(2) g,h : D -> |C are diffable

(3) /\z:-D g'(z) = h'(z)

What can we conclude from this?

(2) g,h : D -> |C are diffable

(3) /\z:-D g'(z) = h'(z)

What can we conclude from this?

\/A:-|C /\z:-D g(z) = h(z) + A

hint:

(g-h)' = g' - h' = 0

g-h is constant

hint:

(g-h)' = g' - h' = 0

g-h is constant

Let F be a s-algebra in X.

A:-F <=> ???

A:-F <=> ???

A:-F <=> 1(A) is F-measurable

1(A) is the characteristic function of A

1(A) : X -> |R*

1(A) is the characteristic function of A

1(A) : X -> |R*

Let F be a s-algebra in X. Let A c X.

1(A) is the characteristic function of A.

1(A) is F-measurable <=> ???

1(A) is the characteristic function of A.

1(A) is F-measurable <=> ???

1(A) is F-measurable <=> A:-F

f : |R -> (-1;1) , f(x) = x / ( |x|+1 )

What is interesting about this function?

What is interesting about this function?

It shows that |R is homeomorphic with (-1;1).

It is even continuously differentiable.

It can be used to define a metric on [-oo,oo].

Then [-oo,oo] is homeorphic with [-1,1].

It is even continuously differentiable.

It can be used to define a metric on [-oo,oo].

Then [-oo,oo] is homeorphic with [-1,1].

Let (X,F,y) be a measure space.

Let f:X->[0,oo] be measurable.

Let A,B:-F and AnB=O.

Prove that Leb*(AuB,f,dy) = ???.

Let f:X->[0,oo] be measurable.

Let A,B:-F and AnB=O.

Prove that Leb*(AuB,f,dy) = ???.

Leb*(AuB,f,dy) = Leb*(A,f,dy) + Leb*(B,f,dy)

page 170 in 1st measure

page 170 in 1st measure

Let F be a s-algebra in X.

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

What can we conclude about the set {x:-X : f(x)>g(x)} ?

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

What can we conclude about the set {x:-X : f(x)>g(x)} ?

{x:-X : f(x)>g(x)} :- F

hint:

{x: f(x)>g(x)} = \\//a:-|Q {x: f(x)>a} n {x: a>g(x)}

page 139 in 1st measure

hint:

{x: f(x)>g(x)} = \\//a:-|Q {x: f(x)>a} n {x: a>g(x)}

page 139 in 1st measure

Let F be a s-algebra in X.

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

What can we conclude about the set {x:-X : f(x)>=g(x)} ?

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

What can we conclude about the set {x:-X : f(x)>=g(x)} ?

{x:-X : f(x)>=g(x)} :- F

hint:

{x: f(x)<g(x)} = \\//a:-|Q {x: f(x)<a} n {x: a<g(x)}

page 139 in 1st measure

hint:

{x: f(x)<g(x)} = \\//a:-|Q {x: f(x)<a} n {x: a<g(x)}

page 139 in 1st measure

Let F be a s-algebra in X.

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

What can we conclude about the set {x:-X : f(x)=g(x)} ?

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

What can we conclude about the set {x:-X : f(x)=g(x)} ?

{x:-X : f(x)=g(x)} :- F

hint:

{x: f(x)<g(x)} = \\//a:-|Q {x: f(x)<a} n {x: a<g(x)}

page 139 in 1st measure

hint:

{x: f(x)<g(x)} = \\//a:-|Q {x: f(x)<a} n {x: a<g(x)}

page 139 in 1st measure

Let F be a s-algebra in X.

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

What are two two ways of proving that f+g is F-measurable?

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

What are two two ways of proving that f+g is F-measurable?

WAY 1:

(1) F : |R^2 -> |R is continuous

(2) h : X -> |R , h(x) = F(f(x),g(x))

thesis: h is F-measurable

Let F(x,y)=x+y (page 138 in 1st measure)

WAY 2:

(-1)*g is measurable,

(1) F : |R^2 -> |R is continuous

(2) h : X -> |R , h(x) = F(f(x),g(x))

thesis: h is F-measurable

Let F(x,y)=x+y (page 138 in 1st measure)

WAY 2:

(-1)*g is measurable,