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 M(y) denote the collection of all y-measurable subsets of |R^p.
Every y-measurable set is a disjoint union of a Borel set
Every y-measurable set is a disjoint union of a Borel set and of a set of measure zero.
page 82 in 2nd measure
Let F be a s-algebra in X.
Let f[n] : X -> |R* be a sequence of measurable functions.
What can we say about this set:
{x:-X : \/A:-|R* lim(n->oo) f[n](x) = A}
it belongs to F
page 83 in 2nd measure
Let F be a s-algebra in X.
Let f[n] : X -> |R* be a sequence of measurable functions.
What can we say about this set:
{x:-X : lim(n->oo) f[n](x) = oo}
it belongs to F
page 83 in 2nd measure
Let F be a s-algebra in X.
Let f[n] : X -> |R* be a sequence of measurable functions.
What can we say about this set:
{x:-X : lim(n->oo) f[n](x) = -oo}
it belongs to F
page 83 in 2nd measure
Let F be a s-algebra in X.
Let f[n] : X -> |R* be a sequence of measurable functions.
Prove that
/\g:-|R {x:-X : lim(n->oo) f[n](x) = g} ???
/\g:-|R {x:-X : lim(n->oo) f[n](x) = g} :- F
page 83 in 2nd measure
Let F be a s-algebra in X.
Let f,g : X -> |R* be measurable.
What can we conclude about function f*g ?
f*g is also measurable
Notice that f,g are allowed to assume the infinite values: oo, - oo.
In the proof, we rely on the theorem that if f,g are measurable and finite,
then their product is also measurable.
Let f : X -> Y. Let W c P(Y).
Let K = { f-1(A) : A:-W }.
Suppose that W is a ring.
What can we conclude about K?
K is also a ring
page 85 in 2nd measure
Let f : X -> Y. Let W c P(Y).
Let K = { f-1(A) : A:-W }.
Suppose that W is a s-ring.
What can we conclude about K?
K is also a s-ring
page 85 in 2nd measure
Let f : X -> Y. Let W c P(Y).
Let K = { f-1(A) : A:-W }.
Suppose that W is an algebra.
What can we conclude about K?
K is also an algebra
page 85 in 2nd measure
Let f : X -> Y. Let W c P(Y).
Let K = { f-1(A) : A:-W }.
Suppose that W is a s-algebra in Y.
What can we conclude about K?
K is a s-algebra in X
page 85 in 2nd measure