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.

What does it mean that (X,F,y) is a measure space in MRW math materials?
(1) X is a non-empty set
(2) F is a s-algebra in X
(3) y : F -> [0,oo]
(4) y is countably additive and y(O)=0
page 102 in 2nd measure
If f : X -> |R*, what does (f+) mean?
(f+) : X -> [0,oo]
/\x:-X (f+)(x) = max( 0, f(x) )
page 168 in 1st measure
page 102 in 2nd measure
If f : X -> |R*, what does (f-) mean?
(f-) : X -> [0,oo]
/\x:-X (f-)(x) = max( 0, -f(x) )
page 168 in 1st measure
page 102 in 2nd measure
Let a,b :- |R*.
Suppose that (a+) = (b+).
What can we conclude?
( a<=0 and b<=0 ) or a=b
page 104 in 2nd measure
Let a,b :- |R*.
Suppose that (a-) = (b-).
What can we conclude?
(a>=0 and b>=0) or a=b
page 104 in 2nd measure
Let a,b :- |R*.
Suppose that (a+)=(b+) and (a-)=(b-).
Is this enough to conclude that a=b ?
YES
page 104 in 2nd measure
Let a,b :- |R*.
Suppose that (a+)!=(b+) or (a-)!=(b-).
What can we conclude from this?
a != b
page 104 in 2nd measure
Let (X,F,y) be a measure space.
Let f,g : X -> |R* be measurable.
Suppose that (f+)~(g+) and (f-)~(g-).
Is this enough to conclude that f~g ?
YES
page 105 in 2nd 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.
(1) Ac|R^p ; /\(n:-|N) A[n]:-MF(y)
(2) lim(n->oo) y*(A[n]+A)=0
A:-MF(y)
page 21 in 2nd measure
(1) f : X -> |R*.
(2) n:-|N
(3) a[1], ..., a[n] :- |R
(4) A[1], ..., A[n] c X , some of them may be empty
(5) X = \\*//(k=1 to n) A[k]
(6) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])(x)
f(X) c { a[1] , ... , a[n] } c |R
f is a simple function
page 248 in 2nd measure