Preface
This i s a seque l t o ou r book s Problems in Mathematical Analysis I,
II (Volume s 4 an d 12 i n th e Studen t Mathematica l Librar y series) .
The boo k deal s with th e Riemann-Stieltje s integra l an d th e Lebesgu e
integral fo r rea l function s o f one rea l variable . Th e boo k i s organize d
in a wa y similar t o tha t o f the firs t tw o volumes , tha t is , i t i s divide d
into tw o parts : problem s an d thei r solutions . Eac h sectio n start s
with a numbe r o f problem s tha t ar e moderat e i n difficulty , bu t som e
of th e problem s ar e actuall y theorems . Thu s i t i s no t a typica l prob -
lem book , bu t rathe r a supplemen t t o undergraduat e an d graduat e
textbooks i n mathematica l analysis . W e hop e tha t thi s boo k wil l b e
of interes t t o undergraduat e students , graduat e students , instructor s
and researche s i n mathematical analysi s an d it s applications. W e also
hope tha t i t wil l b e suitabl e fo r independen t study .
The first chapte r of the book is devoted to Riemann an d Riemann -
Stieltjes integrals . I n Sectio n 1.1w e conside r th e Riemann-Stieltje s
integral wit h respec t t o monotoni c functions , an d i n Sectio n 1.3 w e
turn t o integratio n wit h respec t t o function s o f bounde d variation .
In Sectio n 1.6 w e collec t famou s an d no t s o famou s integra l inequal -
ities. Amon g others , on e ca n fin d OpiaP s inequalit y an d Steffensen' s
inequality. W e clos e th e chapte r wit h th e sectio n entitle d "Jorda n
measure". Th e Jorda n measure , als o calle d conten t b y som e authors ,
vn
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