STUDY OF FUZZY MEASURE AND SOMEPROPERTIES OF NULL-ADDITIVE
FUZZYMEASUREA SummarySubmitted to theKUMAUN UNIVERSITY, NAINITAL,
Uttarakhand, INDIAFOR TE DE!REE OFDoctor of Philosophy in
Mth!"ticsSU"MITTED "YPARUL A#AR$ALM# S$#, M# %hi 'Mathemati$()UNDER
TE SU%ERVISION OFDR% &%S% NAYALA((o$iate %ro*e((orDe+artment o*
mathemati$(!o,t# %o(t-raduate .o&&e-e, Ranikhet
'A&mora)UTTARAKAND#'()*SUMMARY OF TE TESISSTUDY OF FUZZY
MEASURE AND SOMEPROPERTIES OF NULL-ADDITIVE FUZZYMEASURE"YPARUL
A#AR$ALSubmitted to theKUMAUN UNIVERSITY, NAINITAL, Uttarakhand,
INDIAFOR TE DE!REE OFDoctor of Philosophy in Mth!"ticsUNDER TE
SU%ERVISION OFDR% &%S% NAYALA((o$iate %ro*e((orDe+artment o*
mathemati$(!o,t# %o(t-raduate .o&&e-e, Ranikhet
'A&mora)UTTARAKAND#'()*S+""ryThe the(i( entit&ed,,STUDY OF
FUZZY MEASURE AND SOME PROPERTIES OF NULL-ADDITIVE FUZZY MEASURE-i(
di,ided into the *o&&o/in- *i,e
$ha+ter(0.&APTER-)Introdu$tion.&APTER-'%o((ibi&it1
Theor1 ,er(u( %robabi&it1 Theor1 in Fu221 Mea(ure
Theor1.&APTER-/%ro+ertie( o* Nu&&3Additi,e and
Ab(o&ute .ontinuit1 o* a Fu221 Mea(ure.&APTER-0%ro+ertie(
o* Stron- Re-u&arit1 o* Fu221 Mea(ure on Metri$
S+a$e.&APTER-*.ontinuou(, auto3$ontinuou( and$om+&etene((
o* *u221mea(ureFu221(et theor1, de($ribin-*u22ine((
mathemati$a&&1*or the*ir(t time introdu$ed b1 L# A# 4adeh
in 5678, 97:;# In 56th$entur1, mathemati$ian de*ined the $on$e+t(
o* (et( and *un$tion(to re+re(ent a +rob&em# Mea(ure theor1 /a(
de,e&o+ed in(u$$e((i,e (ta-e durin- the &ate 56thand
ear&1 ?;# In thi( $a(ethe uni*ormauto3$ontinuit1 $an be
re+&a$ed /ith the nu&&3additi,e 95:7;# The $on$e+t( o*
t/o (tru$tured $hara$teri(ti$( thatare the +(eudo3metri$ -eneratin-
+ro+ert1 and the auto3$ontinuit1+&a1 im+ortant ro&e( in the
*u221 mea(ure theor1 9>5;#"1meano* thea(1m+toti$(tru$tura&
$hara$teri(ti$( o* *u221mea(ure /e di($u(( the $on,er-en$e in *u221
mea(ure (+a$e and(ome re(u&t( 95:8;, (u$h a( Lebe(-ueA(
theorem, Rie(2A( theorem,E-oro**A(theoremandtheir
-enera&i2ation(, o* *u221mea(uretheor195:7;#
Inthi($ha+ter/e(tudied(ometheorem(onthe$om+&etene(( o*
*u221mea(ure(+a$eand(ome+ro+ertie( o**u221 mea(ure and $on,er-en$e
in mea(ure 95:>;# BIBLIOGRAPHY95; A# DeLu$a, andS# Termini,
'56D), !enera& +ro+ertie( o* entro+1# Re,# Mod#%h1(#, 8:, ),
Amathemati$a& theor1 o*$ommuni$ation# "e&& S1(t# =#
