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 A be a bounded connected subset of |R. Let E>0.
Prove that there exists an open interval G c |R
such that A c G and d(G) <= d(A) + E.
Here d(A) and d(G) denote the length of those intervals.
If A = <a,b>, let G = ]a-E/2 , b+E/2[.
page 108 in 1st measure
Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all intervals in |R^p.
In the context of measure theory, let m be the "volume" function on W.
Let A:-W. Let E>0.
There exists an open G:-W with AcG and m(G)<=m(A)+E.
How can that be proven?
The proof is tedious.
page 109 in 1st measure
Let X be a metric space. Let W c P(X).
Let y : W -> |R*.
How did I choose to define that "y is regular" ?
/\A:-W /\E>0 \/F:-W \/G:-W
F is closed and G is open and F c A c G
and y(G\A)<E and y(A\F)<E
Originally, the item contained:
and y(G)-E <= y(A) <= y(F)+E
Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all elementary subsets of |R^p.
Let m be the "volume" function on W.
Prove that m is regular, that is:
/\A:-W /\E>0 \/F:-W \/G:-W
F is closed and G is open and F c A c G
page 112 in 1st measure
Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all elementary subsets of |R^p.
Let y : W -> [0,oo) be additive and regular.
Extend y into y* so that y* is an outer measure on |R^p.
Prove that it is an outer measure.
Make a wise comment about what's going on.
The proof that y* is an outer measure can be found
on pages 113,116 in 1st measure.
That proof requires neither additivity nor regularity of y.
The proof is conducted in a broader setting on page 95 in 2nd measure.
The proof that y*=y on W requires both additivity
Let 0 <= a[n] <= oo, 0 <= b[n] <= oo.
+(n=1 to n=oo) a[n]+b[n] = ???
+(n=1 to oo) a[n]+b[n] = +(n=1 to oo)_a[n] + ( +(n=1 to oo) b[n] )
Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all elementary subsets of |R^p.
Let y : W -> [0,oo) be additive and regular.
Let y* : P(|R^p) -> [0,oo] be defined as follows:
y*(E) = inf { +(n=1 to n=oo) y(A[n]) : A[n]:-W, A[n] is open, E c U(n:-|N)A[n] }.
/\A:-W y*(A) = y(A)
Recall a more general theorem from which this one follows.
Do not prove the theorem in this item.
page 114 in 1st measure
page 91 in 2nd measure
Let f be a monotonic real-valued function defined on a closed interval in |R.
Does it have to be Riemann-integrable?
Yes.
page 65 in OLDTIMER
Let [a,b] be a bounded interval in |R.
Let f : [a,b] -> |R have both one-sided limits at every point in [a,b].
What can we conclude?
f is bounded,
we exploit the sequential compactness of [a,b]
page 63 in OLDTIMER
Let W be a s-ring in X.
Let y : W -> [0,oo] be countably additive and y(O)=0.
Let A[n] be a sequence of sets in W.
What can we say about the value of y( lim_inf A[n] ) ?
y( lim_inf A[n] ) <= lim_inf y( A[n] )
page 159 in 1st measure