The CPV is usually defined as 'a way to assign values to otherwise undefined integrals.'
Why is it never considered as an integral in its own right that generalizes and extends the Lebesgue integral, in the spirit of how the Lebesgue integral extends the Riemann integral?
No one ever describes the Lebesgue integral as 'a way to assign values to otherwise undefined Riemann integrals.'
Lebesgue integrals can be defined without reference to Riemann integrals. CPV integrals cannot be defined directly. but only as a limit of integrals defined otherwise.