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Consider the p-dimensional Euclidean space |R^p.

Let W be the collection of all intervals in |R^p.

Define an additive function on W

which will eventually lead to the Lebesgue measure on |R^p.

Let W be the collection of all intervals in |R^p.

Define an additive function on W

which will eventually lead to the Lebesgue measure on |R^p.

For -oo < a <= b < oo,

if <a,b> is an interval in |R, then let m(<a,b>) = b - a.

If U = U[1] x ... x U[p], where U[k] is an interval in |R,

then let m(U) = m(U[1]) * ... * m(U[p]).

to see that this function is additive,

if <a,b> is an interval in |R, then let m(<a,b>) = b - a.

If U = U[1] x ... x U[p], where U[k] is an interval in |R,

then let m(U) = m(U[1]) * ... * m(U[p]).

to see that this function is additive,

Consider the p-dimensional Euclidean space |R^p.

In the context of measure theory,

what are the elementary sets?

In the context of measure theory,

what are the elementary sets?

A subset of |R^p is elementary iff

it is a finite union of intervals in |R^p.

Let W be the collection of all intervals in |R^p.

Let E denote the collection of all elementary sets.

Then E = Ws.

page 102 in first measure

it is a finite union of intervals in |R^p.

Let W be the collection of all intervals in |R^p.

Let E denote the collection of all elementary sets.

Then E = Ws.

page 102 in first measure

Consider the p-dimensional Euclidean space |R^p.

In the context of measure theory,

what are the properties of the collection of elementary sets

that follow immediately from its definition?

In the context of measure theory,

what are the properties of the collection of elementary sets

that follow immediately from its definition?

E is a ring of sets.

If W denotes the collection of all intervals in |R^p,

and E denotes the collection of all elementary sets,

then E = Ws = R(W) = the set of all finite disjoint unions of sets from W

If W denotes the collection of all intervals in |R^p,

and E denotes the collection of all elementary sets,

then E = Ws = R(W) = the set of all finite disjoint unions of sets from W

Let A be a complex number.

What can we say about the limit of the sequence {cos(n*A)} ?

What can we say about the limit of the sequence {cos(n*A)} ?

the sequence cos(n*A) does not tend to zero

page 189 in the palace notebook

page 189 in the palace notebook

Let A be an arbitrary set.

Let T be a proposition.

Let Y(a) be a proposition for every a:-A.

Find an equivalent formula for:

[ \/a:-A Y(a) ] => T

Let T be a proposition.

Let Y(a) be a proposition for every a:-A.

Find an equivalent formula for:

[ \/a:-A Y(a) ] => T

[ \/a:-A Y(a) ] => T <=> /\a:-A [ Y(a) => T ]

Let A be an arbitrary set.

Let T be a proposition.

Let Y(a) be a proposition for every a:-A.

Find an equivalent formula for:

/\a:-A [ Y(a) => T ]

Let T be a proposition.

Let Y(a) be a proposition for every a:-A.

Find an equivalent formula for:

/\a:-A [ Y(a) => T ]

/\a:-A [ Y(a) => T ] <=> [ \/a:-A Y(a) ] => T

Let X be a set. Let M be a subset of X.

Let W = { A c X : M c A }.

Describe R(W).

Let W = { A c X : M c A }.

Describe R(W).

R(W) = { E c X : M c E or M n E = O }

page 56 in the 2nd measure notebook

page 56 in the 2nd measure notebook

Let X be a set. Let M be a subset of X.

Let W = { A c X : M c A }.

Describe S(W) - the s-ring generated by W.

Let W = { A c X : M c A }.

Describe S(W) - the s-ring generated by W.

S(W) = { E c X : M c E or M n E = O }

See page 142 in 2nd measure.

(page 57 in the 2nd measure notebook)

See page 142 in 2nd measure.

(page 57 in the 2nd measure notebook)

Let X be a set and let M be a countable subset of X.

Let W = { E c X : M \ E is finite }.

Describe S(W) - the s-ring generated by W.

Let W = { E c X : M \ E is finite }.

Describe S(W) - the s-ring generated by W.

S(W) = P(X)

page 58 in 2nd measure

page 58 in 2nd measure

(1) +(n=1 to n=oo) a[n] = oo

(2) +(n=1 to n=oo) b[n] = b, b is a real number

What can we conclude?

(2) +(n=1 to n=oo) b[n] = b, b is a real number

What can we conclude?

+(n=1 to n=oo) a[n]+b[n] = oo

page 193 in 1st measure

page 193 in 1st measure