Real Analysis III

Lecture 3/31: Measure and Integration on Topological Spaces in General

We construct a chain of concepts in real analysis, beginning with the Borel $\sigma$-algebra $\mathbb{B}$ which is the $\sigma$-algebra on $\mathbb{R}^n$ generated by all open sets.

Following that we have the Lebesgue measure, $m: \mathbb{B} \to [0,+\infty]$, which has lovely properties like countable additivity.

That leads naturally to the Lebesgue integral on $\mathbb{R}^n$ which we consider as a function $S: L^1(\mathbb{R}^d) \to \mathbb{C}$, where $L^1(\mathbb{R}^d)$ is the set of Lebesgue measurable functions on $\mathbb{R}^d$.

Note that $L^1(\mathbb{R}^d)$ is a very permissive class of functions. Contained within it is the set $C_c(\mathbb{R}^d)$, which consists of all continuous functions with compact support. You can generate Riemann integrals on these spaces.

Now in general let $X$ be a locally compact Hausdorff (LCH) space. Since $X$ is a topological space we can take the Borel $\sigma$-algebra, which is of course generated by the open sets on $X$. Since $X$ is LCH this is equal to the sigma-algebra generated by the closed sets, which is equal to the sigma-algebra generated by the compact sets on $X$. By Urysohn’s lemma this is equivalent to the sigma-algebra generated by the continuous functions on $X$ with compact support – this is sometimes called the Baire sigma-algebra.

A Radon measure on $X$ is a non-negative countably additive measure $\mu: \mathbb{B} \to [0, +\infty]$ such that we have

• local finiteness: $\mu(K) < \infty \quad \forall K$ compact
• inner regularity: $\forall \text{ Borel } E, \mu(E) = \sup (\mu(K): K \subset E \text{ compact })$
• outer regularity: $\forall \text{ Borel } E, \mu(E) = \inf (\mu(U): E \subset U \text{ compact })$

Lemma: if $X$ is LCH and $\sigma$-compact, with $\mu$ a Radon measure, then $C_c(X)$ is a dense subset of $L^p(X, \mu) \forall 1 \leq p < \infty$.

As a technical note, we clarify that while $C_c(X)$ is a space of functions (those that are continuous with compact support) while $L^p(X,\mu)$ is a space of functions up to almost-everywhere equivalence. That is, $L^p(X,\mu)$ is a space of equivalence classes of functions, which we elide in the statement of this lemma.

The import of this lemma is that we can approximate any $L^p$ function by a continuous function with compact support. We caveat that $C_c(X)$ is emphatically not dense in $L^{\infty}$.

The above statement corresponds to one of Littlewood’s principles of real analysis: namely that measurable functions are almost continuous. (The other two principles are that measurable sets are almost open sets, and that pointwise convergence is almost uniform convergence).

Here’s another proposition: let $X$ be a compact matric space and $\mu$ a finite nonnegative Borel measure. Then $\mu$ is Radon.

As a demonstration of tactics in this course, let’s prove this. First consider $A$ the set of all Borel subsets that satisfy inner and outer regularity for $\mu$. That is, $A = (E \subset \mathbb{B}: \mu(E) = \sup( \mu(U): U \subset E) = \inf( \mu(U): E \subset U))$.

We want to show that $A = \mathbb{B}$. To start we know that $0, X \in A$. We also know that $A$ is closed under complements and under finite unions and intersections. The last note comes from the inclusion-exclusion principle: consider $E_1, E_2 \in \mathbb{B}$ with nonempty intersection. Then $\mu(E_1) + \mu(E_2) = \sup(\mu(U_1) + \mu(U_2)) = \sup(\mu(U_1 \cup U_2) - \mu(U_1 \cap U_2)) = \mu(E_1 \cup E_2) - \mu(E_1 \cap E_2)$.

Riesz Representation Theorem

Let $X$ be LCH and $\sigma$-compact and let $\mu$ be a Radon measure. This gives us an integration functional $S-d\mu: C_c(X) \to \mathbb{C}$ which is linear and positive, i.e. $\forall f \in C_c(X), f \geq 0 \Rightarrow \int f d\mu \geq 0$.

The Riesz representation theorem provides a converse: let $X$ be LCH and $\sigma$-compact and let $\lambda: C_c(X) \to \mathbb{C}$ be a positive linear functional. Then there exists a unique Radon measure $\mu$ on $X$ such that $\lambda(f) = \int f d\mu$.