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Page 106
The functions totally u - measurable on S , or , if u is understood , totally
measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u -
simple functions . If for every E in { with ulu , E ) < 00 , the product xet off with the
characteristic ...
The functions totally u - measurable on S , or , if u is understood , totally
measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u -
simple functions . If for every E in { with ulu , E ) < 00 , the product xet off with the
characteristic ...
Page 119
( b ) the function g defined by g ( s ) = f ( s ) if s ¢S + US - , g ( s ) = 0 if se S + US - ,
is u - measurable . Next suppose that we consider a function † ( vector or
extended real - valued ) which is defined only on the complement of a p - null set
NCS .
( b ) the function g defined by g ( s ) = f ( s ) if s ¢S + US - , g ( s ) = 0 if se S + US - ,
is u - measurable . Next suppose that we consider a function † ( vector or
extended real - valued ) which is defined only on the complement of a p - null set
NCS .
Page 178
XF ( s ) / ( s ) g ( s ) = lim xe , ( s ) / ( s ) g ( s ) , 8 6 S , and the pointwise limit of a
sequence of measurable functions is measurable , it will suffice then to show that
Xx . fg is measurable . Thus we may and shall assume that v ( u , F ) < 0 and v ( 2
...
XF ( s ) / ( s ) g ( s ) = lim xe , ( s ) / ( s ) g ( s ) , 8 6 S , and the pointwise limit of a
sequence of measurable functions is measurable , it will suffice then to show that
Xx . fg is measurable . Thus we may and shall assume that v ( u , F ) < 0 and v ( 2
...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero