let X be non empty set ; :: thesis: for S being SigmaField of X

for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL

for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let f be PartFunc of X,ExtREAL; :: thesis: for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let B, BF be Element of S; :: thesis: ( f is B -measurable & BF = (dom f) /\ B implies f | B is BF -measurable )

assume that

A1: f is B -measurable and

A2: BF = (dom f) /\ B ; :: thesis: f | B is BF -measurable

for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL

for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let f be PartFunc of X,ExtREAL; :: thesis: for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds

f | B is BF -measurable

let B, BF be Element of S; :: thesis: ( f is B -measurable & BF = (dom f) /\ B implies f | B is BF -measurable )

assume that

A1: f is B -measurable and

A2: BF = (dom f) /\ B ; :: thesis: f | B is BF -measurable

now :: thesis: for r being Real holds BF /\ (less_dom ((f | B),r)) in S

hence
f | B is BF -measurable
by MESFUNC1:def 16; :: thesis: verumlet r be Real; :: thesis: BF /\ (less_dom ((f | B),r)) in S

BF /\ (less_dom ((f | B),r)) c= B /\ (less_dom (f,r)) by A3;

then BF /\ (less_dom ((f | B),r)) = B /\ (less_dom (f,r)) by A5;

hence BF /\ (less_dom ((f | B),r)) in S by A1, MESFUNC1:def 16; :: thesis: verum

end;A3: now :: thesis: for x being object holds

( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) )

then A5:
B /\ (less_dom (f,r)) c= BF /\ (less_dom ((f | B),r))
;( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) )

let x be object ; :: thesis: ( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) )

reconsider xx = x as set by TARSKI:1;

( x in dom (f | B) & ex y being R_eal st

( y = (f | B) . x & y < r ) iff ( x in (dom f) /\ B & ex y being R_eal st

( y = (f | B) . x & y < r ) ) ) by RELAT_1:61;

then A4: ( x in BF & x in less_dom ((f | B),r) iff ( x in B & x in dom f & (f | B) . xx < r ) ) by A2, MESFUNC1:def 11, XBOOLE_0:def 4;

( x in B & x in dom f implies ( f . x < r iff (f | B) . x < r ) ) by FUNCT_1:49;

then ( x in BF /\ (less_dom ((f | B),r)) iff ( x in B & x in less_dom (f,r) ) ) by A4, MESFUNC1:def 11, XBOOLE_0:def 4;

hence ( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) ) by XBOOLE_0:def 4; :: thesis: verum

end;reconsider xx = x as set by TARSKI:1;

( x in dom (f | B) & ex y being R_eal st

( y = (f | B) . x & y < r ) iff ( x in (dom f) /\ B & ex y being R_eal st

( y = (f | B) . x & y < r ) ) ) by RELAT_1:61;

then A4: ( x in BF & x in less_dom ((f | B),r) iff ( x in B & x in dom f & (f | B) . xx < r ) ) by A2, MESFUNC1:def 11, XBOOLE_0:def 4;

( x in B & x in dom f implies ( f . x < r iff (f | B) . x < r ) ) by FUNCT_1:49;

then ( x in BF /\ (less_dom ((f | B),r)) iff ( x in B & x in less_dom (f,r) ) ) by A4, MESFUNC1:def 11, XBOOLE_0:def 4;

hence ( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) ) by XBOOLE_0:def 4; :: thesis: verum

BF /\ (less_dom ((f | B),r)) c= B /\ (less_dom (f,r)) by A3;

then BF /\ (less_dom ((f | B),r)) = B /\ (less_dom (f,r)) by A5;

hence BF /\ (less_dom ((f | B),r)) in S by A1, MESFUNC1:def 16; :: thesis: verum