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