Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Let A,B,M be arbitrary sets.

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

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

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

Let A[n] c X for all natural n.

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

/\x:-X lim_inf 1(A[n]) = ???

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

/\x:-X lim_inf 1(A[n]) = ???

lim_inf 1(A[n]) = 1(lim_inf A[n])

page 11 in 1st measure

page 11 in 1st measure

Let G c |R satisfy:

(1) /\a:-G /\n:-|N n*a :- G

(2) /\a,b:-G a-b :- G

(3) G has a cluster point

What can we conclude about G ?

(1) /\a:-G /\n:-|N n*a :- G

(2) /\a,b:-G a-b :- G

(3) G has a cluster point

What can we conclude about G ?

G is dense in |R

page 70 in OLDTIMER

page 70 in OLDTIMER

Let A be an irrational number.

Consider the set {n*A - [n*A] : n:-|N}.

What can we conclude about this set?

Consider the set {n*A - [n*A] : n:-|N}.

What can we conclude about this set?

It is infinite, hence: it has a cluster point in [0,1].

page 72 in OLDTIMER

page 72 in OLDTIMER

Let A be an irrational number.

Consider the set {n*A + m : n,m are integers}.

What can we conclude about this set?

Consider the set {n*A + m : n,m are integers}.

What can we conclude about this set?

It is dense in |R.

page 72 in OLDTIMER

page 72 in OLDTIMER

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

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

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

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

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

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

Yes.

|R^p = U(n:-|N) [-n,n] x ... x [-n,n]

the Cartesian product of [-n,n] p times

see also page 129 in 1st measure for a different proof

|R^p = U(n:-|N) [-n,n] x ... x [-n,n]

the Cartesian product of [-n,n] p times

see also page 129 in 1st measure for a different proof

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

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

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

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

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

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

Use:

/\E:-M(y) /\e>0 \/G:-M(y) [ G is open and E c G and y*(G\E) < e ]

page 130 in 1st measure

/\E:-M(y) /\e>0 \/G:-M(y) [ G is open and E c G and y*(G\E) < e ]

page 130 in 1st measure

Prove that every open subset of |R^p is a countable union of
bounded open intervals.

page 131 in 1st measure

Prove that every open subset of |R^p is a countable union of
open balls.

a similar proof is on page 131 in 1st measure

Prove that every open subset of |R^p is a countable union of
closed balls.

a similar proof is on page 131 in 1st measure