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a : |N x |N -> [0,oo]

Consider this line:

+(n=1 to n=oo) +(k=1 to k=oo) a(n,k) = +(k=1 to k=oo) +(n=1 to n=oo) a(n,k)

Suppose that infinity is assumed by one of the values pre

Consider this line:

+(n=1 to n=oo) +(k=1 to k=oo) a(n,k) = +(k=1 to k=oo) +(n=1 to n=oo) a(n,k)

Suppose that infinity is assumed by one of the values pre

page 59 in 2nd measure

Determine if it is true:

/\x,y:-|R\{0} |x/y + y/x| >= 2

/\x,y:-|R\{0} |x/y + y/x| >= 2

YES.

Investigate

lim (x,y)->(0,0) x*y / (x + x^2 + y^2)

lim (x,y)->(0,0) x*y / (x + x^2 + y^2)

doesn't exist

smart hint: -y^2 = x

1) x[n] = 1/n, y[n] = 1/n, lim = 0

2) x[n] = -1/n^2, y[n] = -1/n, lim = oo

page 68 in OLDTIMER

smart hint: -y^2 = x

1) x[n] = 1/n, y[n] = 1/n, lim = 0

2) x[n] = -1/n^2, y[n] = -1/n, lim = oo

page 68 in OLDTIMER

Let s be a complex number other than -1.

s / (1+s) = ???

s / (1+s) = ???

s / (1+s) = 1 - 1/(1+s)

Let s be a complex number other than -1.

1 - 1/(1+s) = ???

1 - 1/(1+s) = ???

1 - 1/(1+s) = s / (1+s)

Let y be a nonnegative function on P(X), such that

(1) if A c B c X, then y(A)<=y(B);

(2) if A is a disjoint countable union of sets A[n] c X,

then y(A) <= y(A[1]) + y(A[2]) + ... + y(A[n]) + ...

What further can we conclude about function y?

(1) if A c B c X, then y(A)<=y(B);

(2) if A is a disjoint countable union of sets A[n] c X,

then y(A) <= y(A[1]) + y(A[2]) + ... + y(A[n]) + ...

What further can we conclude about function y?

It is countably subadditive.

page 43 in 2nd measure

page 43 in 2nd measure

Let M,B,A be arbitrary sets.

(A n M) \ (B n M) = ???

(A n M) \ (B n M) = ???

(A n M) \ (B n M) = (A \ B) n M

Let W c P(X) be a ring in X.

Let y : W -> [0,oo] be additive.

y(A u B) <= ???

Let y : W -> [0,oo] be additive.

y(A u B) <= ???

y(A u B) <= y(A) + y(B)

Recall that y(A u B) + y(A n B) = y(A) + y(B).

Notice that y(A n B) >= 0.

Hence y(A u B) <= y(A) + y(B).

page 104 in 1st measure

Recall that y(A u B) + y(A n B) = y(A) + y(B).

Notice that y(A n B) >= 0.

Hence y(A u B) <= y(A) + y(B).

page 104 in 1st measure

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

Prove that there exists a closed interval F c A such that d(A) <= d(F) + E.

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

Prove that there exists a closed interval F c A such that d(A) <= d(F) + E.

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

page 105 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 a closed F:-W with FcA and m(A)<=m(F)+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 a closed F:-W with FcA and m(A)<=m(F)+E.

How can that be proven?

The proof is tedious.

page 106 in 1st measure

page 106 in 1st measure