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Let A be a bounded connected subset of |R. Let E>0.

Prove that there exists an open interval G c |R

such that A c G and d(G) <= d(A) + E.

Here d(A) and d(G) denote the length of those intervals.

Prove that there exists an open interval G c |R

such that A c G and d(G) <= d(A) + E.

Here d(A) and d(G) denote the length of those intervals.

If A = <a,b>, let G = ]a-E/2 , b+E/2[.

page 108 in 1st measure

page 108 in 1st measure

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

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

In the context of measure theory, let m be the "volume" function on W.

Let A:-W. Let E>0.

There exists an open G:-W with AcG and m(G)<=m(A)+E.

How can that be proven?

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

In the context of measure theory, let m be the "volume" function on W.

Let A:-W. Let E>0.

There exists an open G:-W with AcG and m(G)<=m(A)+E.

How can that be proven?

The proof is tedious.

page 109 in 1st measure

page 109 in 1st measure

Let X be a metric space. Let W c P(X).

Let y : W -> |R*.

How did I choose to define that "y is regular" ?

Let y : W -> |R*.

How did I choose to define that "y is regular" ?

/\A:-W /\E>0 \/F:-W \/G:-W

F is closed and G is open and F c A c G

and y(G\A)<E and y(A\F)<E

Originally, the item contained:

and y(G)-E <= y(A) <= y(F)+E

F is closed and G is open and F c A c G

and y(G\A)<E and y(A\F)<E

Originally, the item contained:

and y(G)-E <= y(A) <= y(F)+E

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

Let W be the collection of all elementary subsets of |R^p.

Let m be the "volume" function on W.

Prove that m is regular, that is:

/\A:-W /\E>0 \/F:-W \/G:-W

F is closed and G is open and F c A c G

Let W be the collection of all elementary subsets of |R^p.

Let m be the "volume" function on W.

Prove that m is regular, that is:

/\A:-W /\E>0 \/F:-W \/G:-W

F is closed and G is open and F c A c G

page 112 in 1st measure

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

Let W be the collection of all elementary subsets of |R^p.

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

Extend y into y* so that y* is an outer measure on |R^p.

Prove that it is an outer measure.

Make a wise comment about what's going on.

Let W be the collection of all elementary subsets of |R^p.

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

Extend y into y* so that y* is an outer measure on |R^p.

Prove that it is an outer measure.

Make a wise comment about what's going on.

The proof that y* is an outer measure can be found

on pages 113,116 in 1st measure.

That proof requires neither additivity nor regularity of y.

The proof is conducted in a broader setting on page 95 in 2nd measure.

The proof that y*=y on W requires both additivity

on pages 113,116 in 1st measure.

That proof requires neither additivity nor regularity of y.

The proof is conducted in a broader setting on page 95 in 2nd measure.

The proof that y*=y on W requires both additivity

Let 0 <= a[n] <= oo, 0 <= b[n] <= oo.

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

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

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

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

Let W be the collection of all elementary subsets of |R^p.

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

Let y* : P(|R^p) -> [0,oo] be defined as follows:

y*(E) = inf { +(n=1 to n=oo) y(A[n]) : A[n]:-W, A[n] is open, E c U(n:-|N)A[n] }.

Let W be the collection of all elementary subsets of |R^p.

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

Let y* : P(|R^p) -> [0,oo] be defined as follows:

y*(E) = inf { +(n=1 to n=oo) y(A[n]) : A[n]:-W, A[n] is open, E c U(n:-|N)A[n] }.

/\A:-W y*(A) = y(A)

Recall a more general theorem from which this one follows.

Do not prove the theorem in this item.

page 114 in 1st measure

page 91 in 2nd measure

Recall a more general theorem from which this one follows.

Do not prove the theorem in this item.

page 114 in 1st measure

page 91 in 2nd measure

Let f be a monotonic real-valued function defined on a closed
interval in |R.

Does it have to be Riemann-integrable?

Does it have to be Riemann-integrable?

Yes.

page 65 in OLDTIMER

page 65 in OLDTIMER

Let [a,b] be a bounded interval in |R.

Let f : [a,b] -> |R have both one-sided limits at every point in [a,b].

What can we conclude?

Let f : [a,b] -> |R have both one-sided limits at every point in [a,b].

What can we conclude?

f is bounded,

we exploit the sequential compactness of [a,b]

page 63 in OLDTIMER

we exploit the sequential compactness of [a,b]

page 63 in OLDTIMER

Let W be a s-ring in X.

Let y : W -> [0,oo] be countably additive and y(O)=0.

Let A[n] be a sequence of sets in W.

What can we say about the value of y( lim_inf A[n] ) ?

Let y : W -> [0,oo] be countably additive and y(O)=0.

Let A[n] be a sequence of sets in W.

What can we say about the value of y( lim_inf A[n] ) ?

y( lim_inf A[n] ) <= lim_inf y( A[n] )

page 159 in 1st measure

page 159 in 1st measure