theorem :: MESFUN15:80

for f being PartFunc of REAL,REAL
for a, b being Real
for A being non empty Subset of REAL st ].a,b.[ c= dom f & A = ].a,b.[ & f is_improper_integrable_on a,b & ex c being Real st
( a < c & c < b & abs f is_left_ext_Riemann_integrable_on a,c & abs f is_right_ext_Riemann_integrable_on c,b ) holds
( f | A is_integrable_on L-Meas & improper_integral (f,a,b) = Integral (L-Meas,(f | A)) )