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 Y be an outer measure on X.
Define the corresponding pseudometric on P(X).
Prove that it is a pseudometric.
If A,B c X, then let d(A,B) = Y(A+B).
To prove the triangle inequality, use
A + B c (A + C) u (C + B).
REMARK:
This d can assume plus infinity, hence it is controversial
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
d(A u B , C u D) <= ???
d(A u B , C u D) <= d(A,C) + d(B,D)
page 94 in 1st measure
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
d(A n B , C n D) <= ???
d(A n B , C n D) <= d(A,C) + d(B,D)
page 94 in 1st measure
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
d(A \ B , C \ D) <= ???
d(A \ B , C \ D) <= d(A,C) + d(B,D)
page 94 in 1st measure
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
d(A + B , C + D) <= ???
d(A + B , C + D) <= d(A,C) + d(B,D)
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
In this context,
what does it mean that a sequence of sets converges to a set?
/\n:-|N A[n] c X and A c X
A[n] -> A <=> lim d(A[n],A) = 0 <=> lim Y(A[n] + A) = 0
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
Let A[n] -> A and B[n] -> B.
Prove that A[n] u B[n] -> A u B.
use:
d(A[n] u B[n] , A u B) <= d(A[n],A) + d(B[n],B)
page 95 in 1st measure
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
Let A[n] -> A and B[n] -> B.
Prove that A[n] n B[n] -> A n B.
use:
d(A[n] n B[n] , A n B) <= d(A[n],A) + d(B[n],B)
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
Let A[n] -> A and B[n] -> B.
Prove that A[n] \ B[n] -> A \ B.
use:
d(A[n] \ B[n] , A \ B) <= d(A[n],A) + d(B[n],B)
Let Y be an outer measure on X,
and let d be the corresponding pseudometric on P(X).
Let A[n] -> A and B[n] -> B.
Prove that A[n] + B[n] -> A + B.
use:
d(A[n] + B[n] , A + B) <= d(A[n],A) + d(B[n],B)