# 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 f : X -> Y. Let M c P(X).
Let W = { A c Y : f-1(A) :- M }.
Suppose that M is a s-ideal.
Does W have to be a s-ideal, too?
YES
page 89 in 2nd measure
Let f : X -> Y. Let M c P(X).
Let W = { A c Y : f-1(A) :- M }.
Suppose that M is monotone.
What can we conclude about W ?
W is also monotone
page 89 in 2nd measure
Let f : X -> Y. Let M c P(X).
Let W = { A c Y : f-1(A) :- M }.
Suppose that M is a topology.
Does W also have to be a topology?
YES
page 89 in 2nd measure
Let F be a s-algebra in X.
Let f : X -> |R\{-oo,oo} be measurable.
Let g:|R->|R be such that /\a:-|R {x:-|R : g(x)>a} is Borel.
What can we conclude?
gof is measurable
/\a:-|R {x:-X : g(f(x))>a} :- F
page 90 in 2nd measure
Every bounded sequence of rational numbers contains a Cauchy subsequence.
How can that be proven without using the Dedekind axiom?
Without using the Dedekind axiom, we can prove that every bounded subset of |Q is totally bounded. And every sequence contained in a totally bounded set has a Cauchy subsequence.
page 141 in golden gate
Let f : |R^2 -> |R have both partial derivatives: f1,f2.
(1) /\a,b:-|R |f1(a,b)| <= 2*|b-a|
(2) /\a,b:-|R |f2(a,b)| <= 2*|b-a|
(3) f(0,0) = 0
Prove that |f(5,4)| <= 1.
page 77 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.
Does MF(y) have to be a s-ring?
NO.
We know that if A:-MF(y), then y(A)<oo.
A countable union of elementary sets can be unbounded and have infinite measure.
Let M be a s-algebra in X. Let f : X -> |R* be constant.
Does f have to be M-measurable?
[ \/K:-|R* /\x:-X f(x)=K ] ==> f is measurable
page 136 in 1st measure
Let M be a s-algebra in X.
Let f:X->|R* be M-measurable. Let b:-|R.
What can we conclude about function g:X->|R* defined by g(x)=b*f(x) ?
Function g:X->|R* defined by g(x)=b*f(x) is M-measurable.
page 136 in 1st measure
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.
Prove that for all x:-X
/\x:-X lim_sup 1(A[n])(x) = ???
/\x:-X lim_sup 1(A[n])(x) = 1(lim_sup A[n])(x)
page 11 in 1st measure