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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 M(y) denote the collection of all y-measurable subsets of |R^p.

Every y-measurable set is a disjoint union of a Borel set

Let y : E -> [0,oo) be additive and regular.

Let y* be the corresponding outer measure on |R^p.

Let M(y) denote the collection of all y-measurable subsets of |R^p.

Every y-measurable set is a disjoint union of a Borel set

Every y-measurable set is a disjoint union of a Borel set and
of a set of measure zero.

page 82 in 2nd measure

page 82 in 2nd measure

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

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 : lim(n->oo) f[n](x) = oo}

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

What can we say about this set:

{x:-X : lim(n->oo) f[n](x) = oo}

it belongs to F

page 83 in 2nd measure

page 83 in 2nd measure

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 : lim(n->oo) f[n](x) = -oo}

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

What can we say about this set:

{x:-X : lim(n->oo) f[n](x) = -oo}

it belongs to F

page 83 in 2nd measure

page 83 in 2nd measure

Let F be a s-algebra in X.

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

Prove that

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

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

Prove that

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

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

page 83 in 2nd measure

page 83 in 2nd measure

Let F be a s-algebra in X.

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

What can we conclude about function f*g ?

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

What can we conclude about function f*g ?

f*g is also measurable

Notice that f,g are allowed to assume the infinite values: oo, - oo.

In the proof, we rely on the theorem that if f,g are measurable and finite,

then their product is also measurable.

Notice that f,g are allowed to assume the infinite values: oo, - oo.

In the proof, we rely on the theorem that if f,g are measurable and finite,

then their product is also measurable.

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

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

Suppose that W is a ring.

What can we conclude about K?

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

Suppose that W is a ring.

What can we conclude about K?

K is also a ring

page 85 in 2nd measure

page 85 in 2nd measure

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

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

Suppose that W is a s-ring.

What can we conclude about K?

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

Suppose that W is a s-ring.

What can we conclude about K?

K is also a s-ring

page 85 in 2nd measure

page 85 in 2nd measure

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

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

Suppose that W is an algebra.

What can we conclude about K?

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

Suppose that W is an algebra.

What can we conclude about K?

K is also an algebra

page 85 in 2nd measure

page 85 in 2nd measure

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

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

Suppose that W is a s-algebra in Y.

What can we conclude about K?

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

Suppose that W is a s-algebra in Y.

What can we conclude about K?

K is a s-algebra in X

page 85 in 2nd measure

page 85 in 2nd measure