关于黎曼积分与测度的相关技术疑问

Hey there! Let's work through your questions step by step—this is a really common point of overlap (and confusion) between basic integration and more advanced measure theory, so great call asking about it.

First, that "Riemann measure" you've come across is almost certainly a casual misnomer for the Peano-Jordan measure (often shortened to Jordan measure). The mix-up is totally understandable: since the Riemann integral is built on approximating areas with rectangles, it's tightly linked to how we measure the size of intervals and simple sets, which is exactly what the Jordan measure describes. Jordan measurable sets are those that can be closely approximated by finite unions of rectangles—precisely the kind of sets you're implicitly working with when you calculate upper and lower Riemann sums.

Now, to your core question: Do you need measure theory to formally define the Riemann integral? The straightforward answer is no. You can rigorously build the entire Riemann integral framework using only foundational real analysis tools: things like interval partitions, upper/lower sums, suprema and infima, and limits. This is how it's typically taught in introductory real analysis courses, long before students ever encounter formal measure theory or the Lebesgue integral.

But that quote you mentioned—"using an integral means giving a measure"—hits on a deeper connection that's worth unpacking. Even though you don't need to explicitly use measure theory to define the Riemann integral, the integral still relies on a way to assign "size" to sets (which is exactly what a measure does). When you take a partition of [a,b] into subintervals, you're using the length of each subinterval as your implicit measure. You're weighting the function's value on each subinterval by the size of that interval, which is a measure-like operation, even if you don't label it as such at first.

The big distinction comes when you move beyond Riemann integration: the Jordan measure (tied to Riemann) has limitations—it can't handle sets that are too "fractal" or sparse, like the Cantor set. The Lebesgue integral fixes this by building the integral directly on top of a more general measure (Lebesgue measure), which is why formal measure theory becomes essential for that more powerful framework.

To wrap up with clear takeaways:

• The "Riemann measure" you heard about is almost definitely the Peano-Jordan measure, the implicit size-measure behind Riemann integration.
• You don't need formal measure theory to rigorously define the Riemann integral—basic real analysis suffices.
• The quote about integrals and measures is accurate in the broader sense: every integral relies on some way to assign sizes to sets (a measure), even if that measure is just interval length in the case of the Riemann integral.

备注：内容来源于stack exchange，提问作者Elvis