On Borel and Lebesgue measurability

Walter Rudin’s book is hard to read probably because of two characteristics in his style: he likes to start with great generality, and “sneak in” (well, every now and then) many important details/ideas in lemmas and theorems (even proofs), without explaining the ideas with extra text or example that under which context they are important (again, every now and then). Here I would like to make a comment on measure construction in his book “Real and Complex Analysis” (3 ed, 1986). The following numberings are from this book, unless otherwise stated.

Let $ \mu$ be a Borel measure defined on a sigma algebra $ \mathcal M$ in a general locally compact Hausdorff space. For “any” set $ E \in \mathcal M$, Define “inner regularity” (similar to Theorem 2.14 (d)): $ \mu(E) = \sup {\mu(K): K \subset E, K \mbox{ is compact}}$ and “outer regularity” (Theorem 2.14 (c)): $ \mu(E) = \inf {\mu(V): E \subset V, V \mbox{ is open}}$. In words, outer regularity means we can approximate the measure of a set $ E$ from “above” by open sets, and inner regularity says we can approximate from “below” by compact sets. The great intuition that Lebesgue indicates is that it does not matter whether one approximates the measure from below or above, they are equivalent under the condition that $ E$ is $ \mathbf{\sigma}$-compact (Definition 2.16): “E is a countable union of compact sets”. If a space $ X$ is $ \mathbf{\sigma}$-compact means it is the union of countably many compact subspaces. Hence, with $ \sigma$-compact, one only needs to care about single-sided approximation. This relaxes the more classical definition of integral by Riemann, which requires the upper approximation matches the lower approximation.

The route to construct the measure by Rudin is explained below. He starts with a very general space $ X$ which is 1. Locally Compact + 2. Hausdorff (very different from the others who usually starts from the Euclidean space $ \mathbb{R}^n$) and regard the measures as the functionals on continuous functional space $ C(X)$ by Riesz representation. With Uryson’s lemma, one can characterize the measure’s behavior on the compact and open sets. However, the result of this theorem is a little bit unsatisfactory. That is, the “inner regularity” or the approximation from below can only be shown for sets $ E$ with finite measure $ \mu(E)<\infty$. Apart from this slight drawback, we have essentially the one-sided approximation.

This is why 3. $ \sigma$-compact kicks in for remedy. It is shown in Theorem 2.17 that $ \mu(E)<\infty$ is no longer required, and when $ \mu$ is defined on the Euclidean space, $ \mu$ is called the “Lebesgue measure”, which is a special case of the more general Borel measure on the locally and Hausdorff space. Moreover, in the proof of Theorem 2.20, he shows the Lebesgue integral matches Riemann integral when the integrand is a continuous function.

To conclude, an important message is that Lebesgue measure in its most general sense can be constructed on locally compact, $ \sigma$-compact Hausdorff space, and the traditional Lebesgue measure (I “believe” it is its most original form) on the Euclidean space is a special case.

There are many interesting topics associated with the Lebesgue measure and Borel measure. For example, even on $ \mathbb{R}$ the number of Lebesgue measurable and Borel measurable sets are different. If $ c$ is the cardinality of $ \mathbb{R}$, Borel set is of cardinality of $c$ (can be proved by transfinite induction, implied by the axiom of choice) and Lebesgue measurable sets is of cardinality $2^c$. For example, there exists subset of Cantor sets which are not Borel measurable but Lebesgue measurable. Interestingly, this difference introduces some troubles in defining the weak convergence for stochastic processes, see Billingsley (1968) p. 150-p. 153, or Kosorok’s lecture 8 in UNC.