Measure Zero Sets (Riemann Integration), Real Analysis II

Описание: In this lecture, we make the transition from volume to measure theory, beginning with the concept of measure zero sets. We revisit what it meant, in the earlier framework, for a set to have volume zero, and then explain how measure provides a more flexible and powerful foundation. This shift also clarifies when a function is Riemann integrable over a bounded region in Rn.

In our previous discussion (here    • Volume in Rn with the Characteristic Funct...  ), we defined the volume of a bounded subset of Euclidean space by integrating its characteristic function, provided that function was Riemann integrable. This gave us a working definition of volume but came with limitations, notably the lack of additivity.

After establishing the definition, we prove that the (unbounded) set of rational numbers Q has measure 0. This example illustrates the contrast between volume and measure and prepares us for this fundamental result: Any countable union of measure zero sets has measure zero.

These results give us stronger additivity properties than volume could provide and build toward Lebesgue’s theorem.

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