Notes on Chapter 2 (Positive Borel Measures) of Walter Rudin's Real and Complex Analysis.

Chapter 2

The chapter recaps some basic definitions of vector spaces and linear functionals. It then moves on to a topology recap, in particular:

Hausdorff, compact and locally compact spaces
The vector space of continuous complex functions on $X$ with compact support (note that we will only consider the closure of the set-theoretic support), denoted $C_c(X)$.
We denote by $K\prec f$ that $K$ is a compact subset of $X$, $f\in C_c(X),\ 0\leq f(x)\leq 1$, and $\forall x\in K, f(x) = 1$.
Similarly we denote by $f\prec V$ that $V$ is an open subset of $X$, $f\in C_c(X),\ 0\leq f(x)\leq 1$, and the support of $f$ lies in $V$.

Rudin proceeds to prove the following:

Urysohn's Lemma: Suppose $X$ is a locally compact Hausdorff space, $V$ is open in $X$, $K\subset V$ and $K$ compact. Then there exists $f$ such that $K\prec f\prec V$. □

This version of Urysohn's lemma ends up being used heavily. Using this, we can also construct partitions of unity (on compact $K$, subordinate to a given open covering). Next we spend several pages proving the important Riesz representation theorem in detail:

The Riesz Representation Theorem: Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_c(X)$ (i.e. $f(X)\subset [0,\infty) \Rightarrow \Lambda f\in [0,\infty)$). There exists a $\sigma$-algebra $\mathfrak{M}$ containing all Borel sets of $X$, and a unique positive measure $\mu$ on $\mathfrak{M}$ representing $\Lambda$ in the sense that:

$\Lambda f = \int_X f\ d\mu$ for every $f\in C_c(X)$ (the representation);
$\mu(K) < \infty$ for every compact set $K \subset X$;
For every $E\in\mathfrak{M}$, we have (outer regularity of $\mu$) $$\mu(E) = \inf\{\mu(V): E\subset V, V\text{ open}\};$$
For every open $E$ and every $E\in\mathfrak{M}$ with $\mu(E)<\infty$, we have (not quite inner regularity of $\mu$) $$\mu(E) = \sup\{\mu(K): K\subset E, K\text{ compact}\};$$
If $E\in\mathfrak{M}$, $A\subset E$, and $\mu(E) = 0$, then $A\in\mathfrak{M}$ (Completeness of $(X, \mathfrak{M}, \mu)$ as a measure space, as per chapter 1). □

Essentially, this is more or less a statement about a vector space (positive linear functionals on $C_c(X)$) and its dual. Measurable functions can be approximated by measurable simple functions, which are in turn linear combinations of characteristic functions $\chi_E$ of measurable sets $E$.

Therefore, these characteristic functions form an infinite-dimensional basis for the vector space of measurable simple functions, and the value of the measure $\mu$ is fixed by the value of $\Lambda$ on these characteristic functions. The proof of this version of Riesz in Rudin gives all the details of making this idea work for $C_c(X)$ and proof of all the listed properties.

Now, a measure is regular if every Borel set is regular (both inner and outer regular as above). The notion of $\sigma$-compact sets (countable unions of compact sets) and sets with $\sigma$-finite measure (countable unions of sets with finite measure) are introduced. With some refinement we get the following:

Regularity of measure on $\sigma$-compact spaces: Suppose $X$ is a locally compact, $\sigma$-compact Hausdorff space in the Riesz representation theorem. Then we have the following:

If $E\in\mathfrak{M},\ \varepsilon > 0$, there is a closed $F$ and open $V$ s.t. $F\subset E \subset V$ and $\mu(V-F)<\varepsilon$.
$\mu$ is a regular Borel measure.
If $E\in\mathfrak{M}$, there is an $F_\sigma$ set $A$ and a $G_\delta$ set $B$ s.t. $A\subset E\subset B$ and $\mu(B-A)=0$. □

Corollary: Every $E\in\mathfrak{M}$ is the union of an $F_\sigma$ and a set of measure 0. □

Corollary: Let $X$ be a locally compact Hausdorff space in which every open set is $\sigma$-compact. Let $\lambda$ be any positive Borel measure on $X$ such that $\lambda(K)<\infty$ for every compact $K$. Then $\lambda$ is regular. □

In particular, this last corollary is satisfied by $\mathbb{R}^k$.

Finally, armed with this info, we can construct the Lebesgue measure $m$ (on $\mathbb{R}^k$) as the measure associated to the Riemann integral (which therefore makes the Lebesgue and Riemann integrals agree).

The intuitive 'area'-like properties of the Lebesgue measure are proven, including the determinant property for linear transforms and that measures with most of the important properties are just a constant multiple of the Lebesgue measure.

The Lebesgue-measurable sets end up being exactly those which satisfy the third property in the regularity of measure theorem, which includes all the Borel sets. By a cardinality argument, it can be shown that most Lebesgue-measurable sets are not Borel-measurable. By a separate argument considering $\mathbb{R}$ as a group under addition, and a set of representatives $E$ of cosets of $\mathbb{Q}$ and translates thereof, it can be shown that every set of positive Lebesgue measure must contain some non-measurable subset. (Sets of measure zero have all subsets measurable, by completeness.)

Finally, two useful approximation properties of measurable functions are proven:

Lusin's Theorem: Suppose $f$ is a complex measurable function on locally compact Hausdorff $X$, $\mu(A)<\infty$, $f(x) = 0$ if $x\notin A$, and $\varepsilon > 0$. Then there exists a $g\in C_c(X)$ s.t.

$$\mu(\{x:f(x)\neq g(x)\}) < \varepsilon.$$

Furthermore, we may arrange it so that

$$\sup_{x\in X}|g(x)| \leq \sup_{x\in X}|f(x)|.$$ □

Corollary: Suppose also that $|f|\leq 1$. Then there is a sequence $g_n\in C_c(X),\ |g_n|\leq 1$ and $f(x) = \lim_{n\rightarrow\infty}g_n(x)$ a.e.. □

Plainly put, this theorem lets us approximate bounded measurable $f$ with a sequence of $C_c$ function a.e..

Vitali-Carathéodory Theorem: Suppose $f\in L^1(\mu)$, $f$ is real-valued and $\varepsilon > 0$. Then there exist upper semicontinuous and bounded above $u$ and lower semicontinuous and bounded below $v$ s.t. $u\leq f\leq v$, and

$$\int_X (v-u)\ d\mu < \varepsilon.$$ □

All the proofs in this chapter make heavy use of Urysohn's lemma to get $K\prec f\prec V$ to approximate characteristic functions of various sets. Once the 'sandwiching' properties (inner and outer regularity) of the measures in question are establish, those are often sources of desired $K$ and $V$ which can be used to force $f$ to be sufficiently accurate.

Papa Rudin Chapter 2

⤷jarsp (2022-11-18)

Chapter 2