# 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 W be a s-ring in X.
Let y : W -> [0,oo] be countably additive, y(O)=0.
Define a s-ring in X that contains W.
Don't prove that it is a s-ring, in this item.
(In this SM collection, it will be denoted W^.)
Let W^ = { EcX : \/A,B:-W AcEcB and y(B\A)=0 }.
page 72 in 2nd measure
Let W be a s-ring in X.
Let y : W -> [0,oo] countably additive, y(O)=0.
Prove that N(y^) c W^.
page 73 in 2nd measure
Let F be a s-algebra in X.
Let f : X -> |R* be measurable.
What can we conclude about the sets:
{x:-X : f(x) = oo}, {x:-X : f(x) = -oo} ?
They belong to F.
page 73 in 2nd measure
Let F be a s-algebra in X.
Let f : X -> |R* be measurable.
What can we conclude about the set
{x:-X : -oo < f(x) < oo } ?
It belongs to F.
page 73 in 2nd measure
Let F be a s-algebra in X.
Suppose that F is not equal to P(X).
Prove that there exists a function X -> |R*
that is not measurable.
page 74 in 2nd measure
Let F be a s-algebra in X. Let E:-F.
Let f : X -> |R* be measurable.
Prove that /\a:-|R {x:-E : f(x)>a} ???.
/\a:-|R {x:-E : f(x)>a} :- F
page 74 in 2nd measure
Let F be a s-algebra in X.
Let /\n:-|N f[n] : X - > |R* be measurable.
Let X be a countable disjoint union of sets in F, X = \\//(n:- |N) E[n].
Let f : X -> |R* be defined as follows: x:-E[n] ==> f(x) := f[n](x).
What can we conclude about function f?
It is measurable.
Notice here that X could be written as a finite disjoint union.
page 75 in 2nd measure
page 84 in 2nd measure
Let W be a ring in X. Let y:W->|R* be a additive.
Suppose that whenever {A[n]} is a decreasing sequence of sets in W,
such that its intersection is empty, then lim(n->oo) y(A[n]) = 0.
What can we conclude from this?
y is countably additive on W
page 75 in 2nd measure
Let W be a ring in X. Let y:W->|R* be countably additive.
Let A[n] be a decreasing sequence of sets in W, having empty intersection.
Suppose that y(A[1]) :- |R.
What can we conclude from this?
lim(n->oo) y(A[n]) = 0
page 77 in 2nd measure
Let W be a ring in X. Let y : W -> |R*.
(1) /\B:-W /\AcB [ y(B) = 0 ==> A:-W ]
(2) /\EcX ???
Find an equivalent condition.
(2) /\EcX [ \/A,B:-W AcEcB and y(B\A)=0 ] => E:-W
page 78 in 2nd measure