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

for M being sigma_Measure of S

for E being Element of S

for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

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

for E being Element of S

for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

let M be sigma_Measure of S; :: thesis: for E being Element of S

for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

let E be Element of S; :: thesis: for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

let f be PartFunc of X,ExtREAL; :: thesis: ( E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 implies Integral (M,f) = -infty )

assume that

A1: E = dom f and

A2: f is E -measurable and

A3: f is nonpositive and

A4: M . (E /\ (eq_dom (f,-infty))) <> 0 ; :: thesis: Integral (M,f) = -infty

set g = - f;

A5: E = dom (- f) by A1, MESFUNC1:def 7;

- f = (- 1) (#) f by MESFUNC2:9;

then eq_dom (f,-infty) = eq_dom ((- f),(-infty * (- 1))) by Th9;

then eq_dom (f,-infty) = eq_dom ((- f),+infty) by XXREAL_3:def 5;

then Integral (M,(- f)) = +infty by A1, A2, A3, A4, A5, MESFUNC9:13, MEASUR11:63;

then - (Integral (M,f)) = +infty by A1, A2, Th52;

hence Integral (M,f) = -infty by XXREAL_3:6; :: thesis: verum

for M being sigma_Measure of S

for E being Element of S

for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

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

for E being Element of S

for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

let M be sigma_Measure of S; :: thesis: for E being Element of S

for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

let E be Element of S; :: thesis: for f being PartFunc of X,ExtREAL st E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 holds

Integral (M,f) = -infty

let f be PartFunc of X,ExtREAL; :: thesis: ( E = dom f & f is E -measurable & f is nonpositive & M . (E /\ (eq_dom (f,-infty))) <> 0 implies Integral (M,f) = -infty )

assume that

A1: E = dom f and

A2: f is E -measurable and

A3: f is nonpositive and

A4: M . (E /\ (eq_dom (f,-infty))) <> 0 ; :: thesis: Integral (M,f) = -infty

set g = - f;

A5: E = dom (- f) by A1, MESFUNC1:def 7;

- f = (- 1) (#) f by MESFUNC2:9;

then eq_dom (f,-infty) = eq_dom ((- f),(-infty * (- 1))) by Th9;

then eq_dom (f,-infty) = eq_dom ((- f),+infty) by XXREAL_3:def 5;

then Integral (M,(- f)) = +infty by A1, A2, A3, A4, A5, MESFUNC9:13, MEASUR11:63;

then - (Integral (M,f)) = +infty by A1, A2, Th52;

hence Integral (M,f) = -infty by XXREAL_3:6; :: thesis: verum