[verylongname]

You are just defining the class of Riemann integrable and Lebesgue integrable functions using Jordan measurability. There is nothing wrong in this, but there is nothing different in it either. It is equivalent to the standard limit of simple functions definitions.

[woopwoop]

I agree, I was just pointing out that it is possible to define both the Riemann and Lebesgue integrals by "partitioning the range". The real difference lies in the choice to allow countable rather than finite covers by intervals.