# 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 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 from W.
y( lim_inf A[n] ) < lim_inf y( A[n] )
Is this possible?
Let X = {1,2}. Let y be the counting measure on {1,2}.
Let A[2*n] = {1} and A[2*n+1] = {2}.
Now, y( lim_inf A[n] ) = y( O ) = 0.
But, lim_inf y( A[n] ) = 1.
see page 159 in 1st measure
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 from W.
lim_inf y( A[n] ) < y( lim_inf A[n] )
Is this possible?
NO.
lim_inf y( A[n] ) >= y( lim_inf A[n] )
page 159 in 1st measure
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 from W.
Suppose that y( U{A[n]} ) < oo.
What can we say about the value of lim_sup y( A[n] ) ?
lim_sup y( A[n] ) <= y( lim_sup A[n] )
page 160 in 1st measure
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 from W.
y( lim_sup A[n] ) < lim_sup y( A[n] )
Is this possible?
YES.
Let y be the counting measure on |N.
Let A[n] = {n}. Then lim_sup A[n] = O. Hence y( lim_sup A[n] ) = 0.
But lim_sup y( A[n] ) = 1.
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 from W.
Suppose y( U{A[n]} ) < oo.
y( lim_sup A[n] ) < lim_sup y( A[n] )
NO.
If y( U{A[n]} ) < oo, then
y( lim_sup A[n] ) >= lim_sup y( A[n] ).
page 160 in 1st measure
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 from W.
Suppose that y( U{A[n]} ) < oo.
lim_sup y( A[n] ) < y( lim_sup A[n] )
Let y be the counting measure on {2,3}.
Let A[2*n] = {2}, and A[2*n+1] = {3}.
Now, lim_sup y( A[n] ) = 1.
And, y( lim_sup A[n] ) = y( {2,3} ) = 2.
see page 160 in 1st measure
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 from W.
Suppose that y( U{A[n]} ) < oo.
Let A[n] converge to A.
What can we say about the sequence y( A[n] ) ?
lim(n->oo) y( A[n] ) = y(A)
page 161 in 1st measure
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 from W.
Suppose that y( U{A[n]} ) < oo.
What can we say about the value of y( lim_sup A[n] ) ?
y( lim_sup A[n] ) >= lim_sup y( A[n] )
page 160 in 1st measure
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 from W.
What can we say about the value of lim_inf y( A[n] ) ?
lim_inf y( A[n] ) >= y( lim_inf A[n] )
page 159 in 1st measure
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 from W.
Suppose that y( U{A[n]} ) < oo.
Let A[n] converge to A.
What can we say about the sequence y( A[n] + A ) ?
lim(n->oo) y( A[n] + A ) = 0
page 63 in 2nd measure