let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,REAL
for M being sigma_Measure of S
for F being Finite_Sep_Sequence of S
for a being FinSequence of REAL
for x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
Integral (M,f) = Sum x
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for M being sigma_Measure of S
for F being Finite_Sep_Sequence of S
for a being FinSequence of REAL
for x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
Integral (M,f) = Sum x
let f be PartFunc of X,REAL; :: thesis: for M being sigma_Measure of S
for F being Finite_Sep_Sequence of S
for a being FinSequence of REAL
for x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
Integral (M,f) = Sum x
let M be sigma_Measure of S; :: thesis: for F being Finite_Sep_Sequence of S
for a being FinSequence of REAL
for x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
Integral (M,f) = Sum x
let F be Finite_Sep_Sequence of S; :: thesis: for a being FinSequence of REAL
for x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
Integral (M,f) = Sum x
let a be FinSequence of REAL ; :: thesis: for x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
Integral (M,f) = Sum x
let x be FinSequence of ExtREAL ; :: thesis: ( f is_simple_func_in S & dom f <> {} & f is nonnegative & dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) implies Integral (M,f) = Sum x )
assume that
A1: f is_simple_func_in S and
A2: dom f <> {} and
A3: f is nonnegative and
A4: ( dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) ) and
A5: dom x = dom F and
A6: for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ; :: thesis: Integral (M,f) = Sum x
A7: ( R_EAL f is_simple_func_in S & ( for x being object st x in dom (R_EAL f) holds
0 <= (R_EAL f) . x ) ) by A1, A3, MESFUNC6:49, MESFUNC6:51;
reconsider a0 = a as FinSequence of ExtREAL by MESFUNC3:11;
A8: F,a0 are_Re-presentation_of R_EAL f by A4, MESFUNC3:def 1;
A9: for n being Nat st n in dom x holds
x . n = (a0 . n) * ((M * F) . n) by A6;
thus Integral (M,f) = integral' (M,(R_EAL f)) by A1, A3, MESFUNC6:83
.= integral (M,(R_EAL f)) by A2, MESFUNC5:def 14
.= Sum x by A2, A3, A5, A7, A8, A9, MESFUNC4:3 ; :: thesis: verum