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 be a s-ring in X. Let f : X -> |R*. Let M be dense in |R.
(1) /\a:-M {x:-X : f(x) > a} :- F
What can we conclude?
(2) /\a:-|R {x:-X : f(x) > a} :- F
page 65 in 2nd measure
Let X be a set. Let y : P(X)->[0,oo] , y(O)=0.
M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
Prove that y is additive on M.
Let A:-M and AnB=O.
Then y(AuB) = y( (AuB)nA ) + y( (AuB)\A ).
Notice that y( (AuB)nA ) = y(A).
Since AnB=O, we get y( (AuB)\A ) = y(B).
Hence y(AuB) = y(A) + y(B). This is enough to prove additivity.
page 41 in 2nd measure
Let X be a set. Let y : P(X)->[0,oo] , y(O)=0.
M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
Let E,F belong to M and E n F = O.
Prove that:
/\AcX y( ??? ) = y(A n E) + y(A n F)
/\AcX y( A n (E u F) ) = y(A n E) + y(A n F)
Take any AcX.
Since E:-M, we have y(An(EuF)) = y(An(EuF) n E) + y(An(EuF) \ E).
Since EnF=O, we get y(An(EuF)) = y(A n E) + y(A n F).
page 42 in 2nd measure
Let X be a set. Let y : P(X)->[0,oo] , y(O)=0.
M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
Let E,F belong to M and E n F = O.
Prove that:
/\AcX y( ??? ) = y(A n E n F) + y(A n E n F') + y(A n E' n F)
/\AcX y(A n (E u F) ) = y(A n E n F) + y(A n E n F') + y(A n E' n F')
page 40 in 2nd measure
the line following two stars: (**)
Let X be a set. Let y : P(X)->[0,oo] , y(O)=0.
M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
Let { E[1],E[2],...,E[n] } be a finite disjoint collection of sets in M.
Prove that:
/\AcX y( ??? ) = +(k=1 to k=n) y(A n E[k])
/\AcX y( A n E ) = +(k=1 to k=n) y(A n E[k])
where E = E[1] u E[2] u ... u E[n]
page 43 in 2nd measure
Let y : P(X) -> |R* such that
(1) y(O) = 0
(2) if A c B c X, then y(A)<=y(B)
(3) y is countably subadditive
Let M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
page 44 in 2nd measure
Let y : P(X) -> |R* such that
(1) y(O) = 0
(2) if A c B c X, then y(A)<=y(B)
(3) y is countably subadditive
Let M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
Let { E[n] }(n:-|N) be a disjoint collection of sets in M.
Let E be the union of this collection.
/\AcX y(A n E) = y(A n E[1]) + y(A n E[2]) + ... + y(A n E[n]) + ...
Conclude from this that y is countably additive on M.
page 94, 47 in 2nd measure
Let y:P(X)->|R* be such that
(2) if A c B c X, then y(A)<=y(B)
(3) y is subadditive
Let E be a subset of X such that y(E) = 0.
What can we conclude?
/\AcX y(A) = y(AnE) + y(A\E)
PROOF
A = (A n E) u (A \ E)
y(A) <= y(A n E) + y(A \ E) <= y(E) + y(A) = y(A).
We showed y(A) = y(A n E) + y(A \ E).
page 49 in 2nd measure
Let y be an outer measure on X.
Define the collection of all measurable subsets of X.
State without proof four important theorems about this collection.
Let M = { EcX : /\AcX y(A) = y(AnE) + y(A\E) }.
By definition, E is measurable <=> E:-M.
(1) M is a s-algebra
(2) If { E[n] : n:-|N } is a disjoint collection of sets in M,
and if E is the union of this collection, then
/\AcX y(A n E) = y(A n E[1]) + y(A n E[2]) + ... + y(A n E[n]) + ...
+(n=1 to n=oo) (-1)^(n+1) / n = ???
+(n=1 to n=oo) (-1)^(n+1) / n = log(2)
page 83 in OLDTIMER