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 XX with compact support (note that we will only consider the closure of the set-theoretic support), denoted Cc(X)C_c(X).
  • We denote by KfK\prec f that KK is a compact subset of XX, fCc(X), 0f(x)1f\in C_c(X),\ 0\leq f(x)\leq 1, and xK,f(x)=1\forall x\in K, f(x) = 1.
  • Similarly we denote by fVf\prec V that VV is an open subset of XX, fCc(X), 0f(x)1f\in C_c(X),\ 0\leq f(x)\leq 1, and the support of ff lies in VV.

Rudin proceeds to prove the following:


Urysohn's Lemma: Suppose XX is a locally compact Hausdorff space, VV is open in XX, KVK\subset V and KK compact. Then there exists ff such that KfVK\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 KK, subordinate to a given open covering). Next we spend several pages proving the important Riesz representation theorem in detail:


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

  • Λf=Xf dμ\Lambda f = \int_X f\ d\mu for every fCc(X)f\in C_c(X) (the representation);
  • μ(K)<\mu(K) < \infty for every compact set KXK \subset X;
  • For every EME\in\mathfrak{M}, we have (outer regularity of μ\mu) μ(E)=inf{μ(V):EV,V open};\mu(E) = \inf\{\mu(V): E\subset V, V\text{ open}\};
  • For every open EE and every EME\in\mathfrak{M} with μ(E)<\mu(E)<\infty, we have (not quite inner regularity of μ\mu) μ(E)=sup{μ(K):KE,K compact};\mu(E) = \sup\{\mu(K): K\subset E, K\text{ compact}\};
  • If EME\in\mathfrak{M}, AEA\subset E, and μ(E)=0\mu(E) = 0, then AMA\in\mathfrak{M} (Completeness of (X,M,μ)(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 Cc(X)C_c(X)) and its dual. Measurable functions can be approximated by measurable simple functions, which are in turn linear combinations of characteristic functions χE\chi_E of measurable sets EE.

Therefore, these characteristic functions form an infinite-dimensional basis for the vector space of measurable simple functions, and the value of the measure μ\mu is fixed by the value of Λ\Lambda on these characteristic functions. The proof of this version of Riesz in Rudin gives all the details of making this idea work for Cc(X)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 σ\sigma-compact sets (countable unions of compact sets) and sets with σ\sigma-finite measure (countable unions of sets with finite measure) are introduced. With some refinement we get the following:


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

  • If EM, ε>0E\in\mathfrak{M},\ \varepsilon > 0, there is a closed FF and open VV s.t. FEVF\subset E \subset V and μ(VF)<ε\mu(V-F)<\varepsilon.
  • μ\mu is a regular Borel measure.
  • If EME\in\mathfrak{M}, there is an FσF_\sigma set AA and a GδG_\delta set BB s.t. AEBA\subset E\subset B and μ(BA)=0\mu(B-A)=0.

Corollary: Every EME\in\mathfrak{M} is the union of an FσF_\sigma and a set of measure 0.

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

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


Finally, armed with this info, we can construct the Lebesgue measure mm (on Rk\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 R\mathbb{R} as a group under addition, and a set of representatives EE of cosets of Q\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 ff is a complex measurable function on locally compact Hausdorff XX, μ(A)<\mu(A)<\infty, f(x)=0f(x) = 0 if xAx\notin A, and ε>0\varepsilon > 0. Then there exists a gCc(X)g\in C_c(X) s.t.

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

Furthermore, we may arrange it so that

supxXg(x)supxXf(x).\sup_{x\in X}|g(x)| \leq \sup_{x\in X}|f(x)|.

Corollary: Suppose also that f1|f|\leq 1. Then there is a sequence gnCc(X), gn1g_n\in C_c(X),\ |g_n|\leq 1 and f(x)=limngn(x)f(x) = \lim_{n\rightarrow\infty}g_n(x) a.e..

Plainly put, this theorem lets us approximate bounded measurable ff with a sequence of CcC_c function a.e..


Vitali-Carathéodory Theorem: Suppose fL1(μ)f\in L^1(\mu), ff is real-valued and ε>0\varepsilon > 0. Then there exist upper semicontinuous and bounded above uu and lower semicontinuous and bounded below vv s.t. ufvu\leq f\leq v, and

X(vu) dμ<ε.\int_X (v-u)\ d\mu < \varepsilon.


All the proofs in this chapter make heavy use of Urysohn's lemma to get KfVK\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 KK and VV which can be used to force ff to be sufficiently accurate.