# 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 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.
YES.
Every open subset of |R^p is a countable union of bounded open intervals.
page 131 in 1st 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.
YES.
Every open subset of |R^p is a countable union of bounded open intervals.
Hence M(y) contains all open sets.
And since M(y) is an algebra, it also contains all closed sets.
page 131 in 1st measure
What are Borel sets?
Let (X,G) be a topological space.
The s-algebra generated by G is called the s-algebra of Borel sets.
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.
Yes.
Let G denote the collection of all open subsets of |R^p.
Then G c M(y).
M(y) is a s-algebra, hence s(G) c M(y).
s(G) is the s-algebra generated by G
s(G) is the collection of all Borel sets
Let F be a s-ring in X.
Let f : X -> |R*.
(1) /\a:-|R {x:-X : f(x)>a} :- F
(2) /\a:-|R {x:-X : f(x)>=a} :- F
Prove that (1) => (2).
page 132 in 1st measure
Let F be a s-algebra in X.
Let f : X -> |R*.
(2) /\a:-|R {x:-X : f(x)>=a} :- F
(3) /\a:-|R {x:-X : f(x)<a} :- F
Prove that (2) => (3).
page 132 in 1st measure
Let F be a s-ring in X.
Let f : X -> |R*.
(3) /\a:-|R {x:-X : f(x)<a} :- F
(4) /\a:-|R {x:-X : f(x)<=a} :- F
Prove that (3)=>(4).
page 132 in 1st measure
Let F be a s-algebra in X.
Let f : X -> |R*.
(4) /\a:-|R {x:-X : f(x)<=a} :- F
(1) /\a:-|R {x:-X : f(x)>a} :- F
Prove that (4)=>(1).
page 132 in 1st measure
Let F be a s-algebra in X.
Let f : X -> |R*.
What does it mean that "f is measurable" ?
(1) /\a:-|R {x:-X : f(x)>a} :- F
(2) /\a:-|R {x:-X : f(x)>=a} :- F
(3) /\a:-|R {x:-X : f(x)<a} :- F
(4) /\a:-|R {x:-X : f(x)<=a} :- F
these four are equivalent
page 132,133 in 1st measure
Let F be a s-algebra in X.
Let f : X -> |R* be measurable.
What can we conclude about function |f| ?
|f| is measurable
page 133 in 1st measure