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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 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 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?

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

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] ) ?

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

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?

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 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] )

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

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 : 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 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] ) ?

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

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] ) ?

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

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] ) ?

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

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 ) ?

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

page 63 in 2nd measure