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 T be a proposition. Let A be a set.
Let Y(a) be a proposition for every a:-A.
Find an equivalent condition for:
(1) T => [ /\a:-A Y(a) ].
(2) /\a:-A [ T => Y(a) ]
Let T be a proposition. Let A be a set.
Let Y(a) be a proposition for every a:-A.
Find an equivalent condition for:
(1) /\a:-A [ T => Y(a) ]
(2) T => [ /\a:-A Y(a) ]
Let W be a s-ring in X.
Let y : W -> [0,oo] be countably additive, y(O)=0.
Prove that N(y) c W <=> ???.
N(y) c W <=> W^ = W
In the proof use a theorem which works in a simpler setting.
Simply recall that theorem.
page 78 in 2nd measure
Let W be a ring in X. Let y : W -> |R*.
(1) /\EcX [ \/A,B:-W AcEcB and y(B\A)=0 ] => E:-W
(2) /\B:-W ???
Find an equivalent condition.
(2) /\B:-W /\AcB [ y(B) = 0 ==> A:-W ]
page 78 in 2nd measure
Let W be a s-ring in X.
Let y : W -> [0,oo] countably additive, y(O)=0.
Prove that W^ = W <=> ???.
W^ = W <=> N(y) c W
In the proof use a theorem which works in a simpler setting.
Simply recall that theorem.
page 78 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.
Let A c |R^p and y*(A) = 0.
A :- MF(y)
page 79 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.
Let M(y) denote the collection of all y-measurable subsets of |R^p.
For every A:-M(y) there exist two Borel sets F,G
such that F c A c G and y(G\F)=y(G\A)=y(A\F)=0.
Hence M(y) c B^, if B denotes the set of all Borel sets
and B^ as on page 72 in 2nd measure.
page 81 in 2nd measure
Let X be a complete metric space. Let W be a ring in X.
Let y : W -> [0,oo) be additive and regular.
Let A:-W and suppose that A is totally bounded.
How can we express y(A) in a very interesting way?
Prove the answer.
y(A) = inf { +(n=1 to oo) y(E[n]) : E[n] is an open subset of W and A c \\//E[n] }.
page 91 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 M(y) denote the collection of all y-measurable subsets of |R^p.
Let |B denote the set of all Borel subsets of |R^p.
Prove that
page 81 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 M(y) denote the collection of all y-measurable subsets of |R^p.
Every y-measurable set is a disjoint union of _______ and
Every y-measurable set is a disjoint union of a Borel set and of a set of measure zero.
page 82 in 2nd measure