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 Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

Let B,A c X and Y(A) < oo , Y(B) < oo.

|Y(A) - Y(B)| <= ???

and let d be the corresponding pseudometric on P(X).

Let B,A c X and Y(A) < oo , Y(B) < oo.

|Y(A) - Y(B)| <= ???

|Y(A) - Y(B)| <= d(A,B)

|Y(A) - Y(B)| <= Y(A+B)

way (1): use d(A,O) <= d(A,B) + d(B,O).

way (2): use Y(A) <= Y(AuB) <= Y(A+B) + Y(AnB) <= Y(A+B) + Y(B)

|Y(A) - Y(B)| <= Y(A+B)

way (1): use d(A,O) <= d(A,B) + d(B,O).

way (2): use Y(A) <= Y(AuB) <= Y(A+B) + Y(AnB) <= Y(A+B) + Y(B)

Let Y be an outer measure on X,

and let d be the corresponding pseudometric on P(X).

Suppose that /\n:-|N Y(A[n]) < oo , Y(A) < oo.

Suppose that A[n] -> A, ( lim(n->oo) Y(A[n]+A) = 0 ).

What can we conclude?

and let d be the corresponding pseudometric on P(X).

Suppose that /\n:-|N Y(A[n]) < oo , Y(A) < oo.

Suppose that A[n] -> A, ( lim(n->oo) Y(A[n]+A) = 0 ).

What can we conclude?

lim Y(A[n]) = Y(A)

hint: use |Y(A[n]) - Y(A)| <= d(A[n],A)

page 96 in 1st measure

hint: use |Y(A[n]) - Y(A)| <= d(A[n],A)

page 96 in 1st measure

Let a : |N x |N -> [0,oo].

Consider this line:

+(n=1 to n=oo) +(k=1 to k=oo) a(n,k) = +(k=1 to k=oo) +(n=1 to n=oo) a(n,k)

Prove this line supposing that infinity does not occur in

Consider this line:

+(n=1 to n=oo) +(k=1 to k=oo) a(n,k) = +(k=1 to k=oo) +(n=1 to n=oo) a(n,k)

Prove this line supposing that infinity does not occur in

page 59 in 2nd measure

page 97 in 1st measure

page 97 in 1st measure

Let X[n] be a sequence of sets. Let X be the union of all X[n].

Let S[n] be a s-ring in X[n], for every n:-|N.

Let S = { E c X : /\n:-|N ( E n X[n] :- S[n] ) }.

What can we conclude about S?

Let S[n] be a s-ring in X[n], for every n:-|N.

Let S = { E c X : /\n:-|N ( E n X[n] :- S[n] ) }.

What can we conclude about S?

S is a s-ring

page 99 in 1st measure

page 99 in 1st measure

Let X[n] be a sequence of sets. Let X be the union of all X[n].

Let S[n] be a s-ring in X[n], for every n:-|N.

Let y[n] : S[n] -> [0,oo] be countably additive, for every n:-|N.

Let S = { E c X : /\n:-|N ( E n X[n] :- S[n] ) }.

Let Y : S -> [0,oo] be defined by Y(E) = +(n=1 to n=oo) y[n](E n X[n]).

Let S[n] be a s-ring in X[n], for every n:-|N.

Let y[n] : S[n] -> [0,oo] be countably additive, for every n:-|N.

Let S = { E c X : /\n:-|N ( E n X[n] :- S[n] ) }.

Let Y : S -> [0,oo] be defined by Y(E) = +(n=1 to n=oo) y[n](E n X[n]).

ANSWER: Y is countably additive.

page 99 in 1st measure

By the way, S is a s-ring in X.

But that is the subject matter of the previous item.

page 99 in 1st measure

By the way, S is a s-ring in X.

But that is the subject matter of the previous item.

Consider the set of all bounded intervals on the real line,
including singletons and the empty set. Is it a ring?

NO.

[1,7] \ [2,6] does not belong

[1,7] \ [2,6] does not belong

Consider the set of all bounded intervals on the real line,
including singletons and the empty set. Is it a semi-ring?

Yes.

page 60 in 1st measure

page 60 in 1st measure

Consider the p-dimensional Euclidean space |R^p.

In the context of measure theory,

how do we define the term "interval" in |R^p ?

In the context of measure theory,

how do we define the term "interval" in |R^p ?

An interval in |R is a bounded connected subset of |R,
including the empty set.

If {P[n]}(n=1 to n=p) is a collection of intervals in |R,

then the Cartesian product P[1] x P[2] x ... x P[p] is an interval in |R^p.

If {P[n]}(n=1 to n=p) is a collection of intervals in |R,

then the Cartesian product P[1] x P[2] x ... x P[p] is an interval in |R^p.

Consider the p-dimensional Euclidean space |R^p.

In the context of measure theory,

consider the collection of all intervals in |R^p.

Is it a ring?

In the context of measure theory,

consider the collection of all intervals in |R^p.

Is it a ring?

NO.

[1,7] \ [2,4] does not belong

[1,7] \ [2,4] does not belong

Consider the p-dimensional Euclidean space |R^p.

In the context of measure theory,

consider the collection of all intervals in |R^p.

Is it a semi-ring?

In the context of measure theory,

consider the collection of all intervals in |R^p.

Is it a semi-ring?

Yes.

Recall that the intervals in |R form a semi-ring.

Then recall that if X,Y are semi-rings,

then { A x B : A:-X, B:-Y } is also a semi-ring.

page 101 in 1st measure

Recall that the intervals in |R form a semi-ring.

Then recall that if X,Y are semi-rings,

then { A x B : A:-X, B:-Y } is also a semi-ring.

page 101 in 1st measure