:: MESFUNC4 semantic presentation

theorem Th1: :: MESFUNC4:1

for F, G, H being FinSequence of ExtREAL st ( for i being Nat st i in dom F holds
0. <=' F . i ) & ( for i being Nat st i in dom G holds
0. <=' G . i ) & dom F = dom G & H = F + G holds
Sum H = (Sum F) + (Sum G)

proof end;

theorem Th2: :: MESFUNC4:2

for X being non empty set
for S being sigma_Field_Subset of X
for M being sigma_Measure of S
for n being Nat
for f being PartFunc of X, ExtREAL
for F being Finite_Sep_Sequence of S
for a, x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & ( for x being set st x in dom f holds
0. <=' f . x ) & F,a are_Re-presentation_of f & dom x = dom F & ( for i being Nat st i in dom x holds
x . i = (a . i) * ((M * F) . i) ) & len F = n holds
integral X,S,M,f = Sum x

proof end;

theorem Th3: :: MESFUNC4:3

for X being non empty set
for S being sigma_Field_Subset of X
for f being PartFunc of X, ExtREAL
for M being sigma_Measure of S
for F being Finite_Sep_Sequence of S
for a, x being FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} & ( for x being set st x in dom f holds
0. <=' f . x ) & F,a are_Re-presentation_of f & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) holds
integral X,S,M,f = Sum x

proof end;

theorem Th4: :: MESFUNC4:4

for X being non empty set
for S being sigma_Field_Subset of X
for f being PartFunc of X, ExtREAL
for M being sigma_Measure of S st f is_simple_func_in S & dom f <> {} & ( for x being set st x in dom f holds
0. <=' f . x ) holds
ex F being Finite_Sep_Sequence of S ex a, x being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) & integral X,S,M,f = Sum x )

proof end;

theorem :: MESFUNC4:5

for X being non empty set
for S being sigma_Field_Subset of X
for M being sigma_Measure of S
for f, g being PartFunc of X, ExtREAL st f is_simple_func_in S & dom f <> {} & ( for x being set st x in dom f holds
0. <=' f . x ) & g is_simple_func_in S & dom g = dom f & ( for x being set st x in dom g holds
0. <=' g . x ) holds
( f + g is_simple_func_in S & dom (f + g) <> {} & ( for x being set st x in dom (f + g) holds
0. <=' (f + g) . x ) & integral X,S,M,(f + g) = (integral X,S,M,f) + (integral X,S,M,g) )

proof end;

theorem :: MESFUNC4:6

for X being non empty set
for S being sigma_Field_Subset of X
for M being sigma_Measure of S
for f, g being PartFunc of X, ExtREAL
for c being R_eal st f is_simple_func_in S & dom f <> {} & ( for x being set st x in dom f holds
0. <=' f . x ) & 0. <=' c & c <' +infty & dom g = dom f & ( for x being set st x in dom g holds
g . x = c * (f . x) ) holds
integral X,S,M,g = c * (integral X,S,M,f)

proof end;