# Math ASCII Notation Demo

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 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.
Prove that y* is additive on MF(y).
page 119 in 1st measure
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.
Recall that MF(y) is a ring.
Use the fact that every countable union of sets in a ring can be written as a disjoint countable union of sets in that ring.
page 120 in 1st measure
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.
y*(A) = +(n=1 to n=oo) y*(A[n])
Recall that MF(y) is a ring and y* is additive on MF(y)
and use this in the proof.
page 121 in 1st measure
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 from W.
y( lim_inf A[n] ) < lim_inf y( A[n] ) < lim_sup y( A[n] ) < y( lim_sup A[n] )
Let X = {1,2,3}.
Let y be the counting measure on X.
Let A[2n] = {1}, A[2n+1] = {2,3}.
We have 0 < 1 < 2 < 3.
see pages 159,160 in 1st measure
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.
y*(A) < oo
PROOF:
Since A:-MF(y), there is B:-E such that y*(A+B)<7, and y*(B)<oo.
Notice that A c (A+B) u B. Hence y*(A) <= y*(A+B) + y*(B) < oo.
page 122 in 1st measure
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.
A :- MF(y)
For the proof, use the fact below.
If A is a countable disjoint union of sets in MF(y), say A = U(n:-|N)_A[n],
then y*(A) = +(n:-|N)_y*(A[n]).
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.
YES.
In order to prove this, we use:
(1) MF(y) is a ring
(2) A:-M(y) ==> [ y*(A)<oo <=> A:-MF(y) ]
page 124 in 1st measure
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.
Use the fact that y* is countably additive on MF(y).
page 125 in 1st measure
Take any A[n],A :- M(y) such that \\*//(n:-|N)_A[n] = A.
If \/k:-|N y(A[k])=oo, then y(A) = oo = +(n=1 to oo) y(A[n]).
If /\k:-|N y(A[k])<oo, then /\k:-|N A[k] :- MF(y)
and use Theorem 86 on page 121 in 1st measure.
( U(t:-T) G[t] ) \ ( U(t:-T) E[t] ) c ???
( U(t:-T) G[t] ) \ ( U(t:-T) E[t] ) c U(t:-T) G[t]\E[t]
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.
page 127 in 1st measure
page 79 in 2nd measure