:: deftheorem DefXvol defines X-vol MEASUR11:def 14 :
for X1, X2 being non empty set
for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite holds
for b₇ being V102() Function of X2,ExtREAL holds
( b₇ = X-vol (E,M1) iff ( ( for y being Element of X2 holds b₇ . y = M1 . (Measurable-Y-section (E,y)) ) & ( for V being Element of S2 holds b₇ is_measurable_on V ) ) );