[woopwoop]

This is a nice summary, but I've always found the characterization of Riemann integrals as "partitioning the domain" and Lebesgue integrals as "partitioning the range" unsatisfying. This is mainly an artifice of the common constructions, but one can give definitions of the Riemann and Lebesgue integrals where the only difference is that, in several places, one must replace the word "finite" with the word "countable". Here is one such:

The definition of the length of an interval should be obvious. Let I be an interval, and let E be a subset of I. Define its Jordan outer measure to be the inf of the sums of the lengths of finite collections of intervals covering E. Define its Jordan inner measure to be the length of I minus the Jordan outer measure of I \setminus E. E is called Jordan measurable if its outer and inner Jordan measures are equal. A function s is Jordan simple if it is a linear combination of characteristic functions of Jordan measurable sets. Define the integral of Jordan simple functions in the obvious way. A bounded function on I is Riemann integrable if and only if it is the uniform limit of Jordan simple functions, and its Riemann integral is the limit of the integrals of the approximating simple functions.

If, in the previous paragraph, one replaces the word "finite" with the word "countable", and the names "Jordan" and "Riemann" with "Lebesgue", one recovers the Lebesgue integral.

[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.