## Banach Measures

If you have never read one of Elias Stein’s books, you are missing out. “Harmonic Analysis: Real-Variables, Orthogonality, and Oscillatory” and “Introduction to Fourier Analysis on Euclidean Spaces” (co-authored by Guido Weiss) are not light bedtime reading, but they make the reader–or at least, me–get excited about Analysis. Moreover, I increasingly find the benefits of reading the masters when trying to learn something. Today, I want to share a result from Book 4 of the four-volume “Princeton Lectures in Analysis” geared towards undergraduates and beginning graduate students.

We will show that there exists a finitely-additive measure ${\hat{m}}$ defined on all subsets of ${\mathbb{R}^{d}}$ that is translation-invariant and agrees with the Lebesgue measure on the Lebesgue ${\sigma}$-algebra. Of course, such a measure has no hope of being countably additive (${\sigma}$-additive) as we know from the existence of Vitali sets. Such a measure ${\hat{m}}$ is called a Banach measure.

Theorem 1 There exists an extended-valued nonnegative function ${\hat{m}:\mathcal{P}(\mathbb{R}^{d})\rightarrow[0,\infty]}$ satisfying

(Finitely Additive) ${\hat{m}(E_{1}\cup E_{2})=\hat{m}(E_{1})+\hat{m}(E_{2})}$, if ${E_{1},E_{2}\subset\mathbb{R}^{d}}$ are disjoint;

(Agreement) ${\hat{m}(E)=m(E)}$ if ${E\subset\mathbb{R}^{d}}$ is Lebesgue measurable;

(Translation Invariance) ${\hat{m}(E+h)=\hat{m}(E)}$ for all ${E\subset\mathbb{R}^{d}}$ and ${h\in\mathbb{R}^{d}}$.

To prove Theorem 1, we will follow the functional analytic approach to measure and integration by first defining an integral as a linear functional on a suitable space of functions and then defining a measure by evaluating the integral of characteristic functions. Specifically, we use the Hahn-Banach theorem to extend the Lebesgue integral to the space of bounded (possibly nonmeasurable) functions on the ${d}$-torus ${\mathbb{R}^{d}/\mathbb{Z}^{d}}$.

A quick remark on notation: we denote the power set and characteristic function of a set ${X}$ by ${\mathcal{P}(X)}$ and ${\chi_{E}}$, respectively.

Theorem 2 There exists a linear functional ${f\mapsto I(f)}$ defined on all bounded functions on ${\mathbb{R^{d}}/\mathbb{Z^{d}}}$ such that(Positive Semidefinite) ${I(f)\geq 0}$, if ${f\geq 0}$;

(Linearity) ${I(\alpha f_{1}+\beta f_{2})=\alpha I(f_{1})+\beta I(f_{2})}$ for all ${f_{1}, f_{2}}$ and ${\alpha,\beta\in\mathbb{R}}$;

(Agreement) ${I(f)=\int_{[0,1]^{d}}f(x)dx}$, if ${f}$ is Lebesgue measurable;

(Translation Invariance) ${I(f_{h})=I(f)}$, where ${f_{h}(x):=f(x-h)}$, for all ${h\in\mathbb{R}^{d}}$.

Proof: Let ${V}$ denote the real vector space of bounded functions ${f:\mathbb{R}^{d}/\mathbb{Z}^{d}}$. Let ${V_{0}}$ denote the subspace of Lebesgue measurable functions. Denote the functional on ${V_{0}}$ defined by Lebesgue integration by ${I_{0}}$. To use the Hahn-Banach theorem, we need to find a suitable seminorm ${p}$ such that ${I_{0}(f)\leq p(f)}$ for all ${f\in V_{0}}$. The clever construction of ${p}$ is due to Polish mathematician Stefan Banach.

Let ${A=\left\{a_{1},\ldots,a_{N}\right\}}$ be a sequence in ${\mathbb{R}^{d}}$ of length ${N}$. Given ${A}$, define

$\displaystyle M_{A}(f):=\sup_{x\in\mathbb{R}^{d}}\left(\dfrac{1}{N}\sum_{j=1}^{N}f(x+a_{j})\right)\in\mathbb{R}.$

We define ${p(f):=\inf_{A}\left\{M_{A}(f)\right\}}$, where the infimum is taken over all finite sequencess ${A}$ of arbitrary length. The definition of ${p(f)}$ should remind you of upper Darboux sums used in Riemann integration. It is clear that ${p}$ is well-defined, as ${f}$ is bounded.

I claim that ${p}$ is a sublinear functional on ${V}$. Indeed, it is evident that ${p}$ is positively homgeneous. Given ${\varepsilon>0}$, we can find find sequences ${A}$ and ${B}$ of length ${N_{1}}$ and ${N_{2}}$, respectively, such that ${M_{A}(f_{1})\leq p(f_{1})+\varepsilon}$ and ${M_{A}(f_{2})\leq p(f_{2})+\varepsilon}$. Define a sequence ${C:=\left\{a_{i}+b_{j}\right\}_{1\leq i\leq N_{1},1\leq j\leq N_{2}}}$ of length ${N_{1}\cdot N_{2}}$. It is evident that ${M_{C}(f_{1}+f_{2})\leq M_{C}(f_{1})+M_{C}(f_{2})}$. Since ${M_{E}(f)=M_{E'}(f_{h})}$, where any ${h\in\mathbb{R}^{d}}$ and ${E'}$ is a translate of ${E}$, we see that

$\displaystyle\dfrac{1}{N_{2}}\sum_{j=1}^{N_{2}}\dfrac{1}{N_{1}}\sum_{i=1}^{N_{1}}f_{1}(x+a_{i}+b_{j})\leq\dfrac{1}{N_{2}}\sum_{j=1}^{N_{2}}M_{A-b_{j}}(f_{1})=M_{A}(f_{1}).$

Taking the supremum of the LHS over all ${x\in\mathbb{R}^{d}}$, we see that ${M_{C}(f_{1})\leq M_{A}(f_{1})}$. A completely analogous argument shows that ${M_{C}(f_{2})\leq M_{C}(f_{2})}$. Putting these inequalities together, we conclude that

$\begin{array}{lcl}\displaystyle p(f_{1}+f_{2})&\leq&\displaystyle M_{C}(f_{1}+f_{2})\\[1.5 em]\displaystyle&=&\displaystyle\leq M_{C}(f_{1})+M_{C}(f_{2})\\[1.5 em]&\leq&\displaystyle M_{A}(f_{1})+M_{B}(f_{2})\\[1.5 em]&\leq&\displaystyle p(f_{1})+p(f_{2})+2\varepsilon.\end{array}$

Letting ${\varepsilon\downarrow 0}$ completes the proof of sublinearity.
I claim that ${I_{0}(f)\leq p(f)}$ for all ${f\in V_{0}}$. Indeed, by translation invariance and linearity,

$\displaystyle I_{0}(f)=\dfrac{1}{N}\sum_{i=1}^{N}\int_{[0,1]^{d}}f(x+a_{i})dx=\int_{[0,1]^{d}}\left(\dfrac{1}{N}\sum_{i=1}^{N}f(x+a_{i})\right)dx\leq M_{A}(f)$

for all finite sequences ${A\subset\mathbb{R}^{d}}$. Taking the infimum of the RHS over all such ${A}$, we conclude that ${I_{0}(f)\leq p(f)}$.

We apply the Hahn-Banach theorem to ${I_{0}}$ to obtain a linear functional ${I:V\rightarrow\mathbb{R}}$ bounded by ${p}$. Properties (ii) and (iii) have already been proven. For (i), observe that if ${f\leq 0}$, then ${M_{A}(f)\leq 0}$ for all sequencess ${A}$, whence ${p(f)\leq 0}$. If ${f\leq 0}$, then applying this observation to ${-f}$ shows that ${I(f)\geq 0}$. To establish (iv), it suffices by symmetry to show that ${p(f-f_{h})\leq 0}$, where ${h\in\mathbb{R}^{d}}$ is fixed. Consider the set ${A:=\left\{h,2h,\ldots,Nh\right\}}$, for some integer ${N\geq 1}$. Then

$\displaystyle \begin{array}{lcl}\displaystyle\dfrac{1}{N}\sum_{j=1}^{N}(f-f_{h})(x+jh)&=&\displaystyle\dfrac{1}{N}\sum_{j=1}^{N}\left[f(x+jh)-f(x+(j-1)h)\right]\\[2 em]&=&\displaystyle\dfrac{1}{N}\left[f(x+Nh)-f(x)\right]\\[2 em]&=&\displaystyle\dfrac{2\left\|f\right\|_{\infty}}{N}\end{array}$

Letting ${N\uparrow\infty}$, we obtain that ${p(f-f_{h})\leq M_{A}(f-f_{h})\leq 0}$. $\Box$

By defining ${\hat{m}:\mathcal{P}(\mathbb{R}^{d}/\mathbb{Z}^{d})\rightarrow[0,\infty]}$ by ${\hat{m}(E):=I(\chi_{E})}$, we obtain the following corollary.

Corollary 3 There exists a nonnegative set function ${\mathcal{P}(\mathbb{R}^{d}/\mathbb{Z}^{d})\rightarrow[0,\infty]}$ such that

(Finite Additivity) ${\hat{m}(E_{1}\cup E_{2})=\hat{m}(E_{1})+\hat{m}(E_{2})}$, if ${E_{1},E_{2}\subset\mathbb{R}^{d}/\mathbb{Z}^{d}}$ are disjoint;

(Agreement) ${\hat{m}(E)=m(E)}$, if ${E}$ is Lebesgue measurable;

(Translation Invariance) ${\hat{m}(E+h)=\hat{m}(E)}$ for all ${h\in\mathbb{R}^{d}}$.

We are now ready to prove Theorem 1. The plan is to partition ${\mathbb{R}^{d}}$ into countably many disjoint cubes that can be translated to the unit cube ${Q_{0}:=[0,1)^{d}}$ by an element of ${\mathbb{Z}^{d}}$. We first define ${\hat{m}}$ locally on disjoint cubes by computing ${\hat{m}_{0}}$ on the translate to ${Q_{0}}$. For an arbitrary set ${E}$, we define ${\hat{m}(E)}$ as the “infinite sum” of these local quantities.

Proof of Theorem 1: Let ${Q_{n}=Q_{0}+n}$ denote the denote the translate of the unit cube by ${n=(n_{1},\ldots,n_{d})\in\mathbb{Z}^{d}}$. The collection ${\left\{Q_{n}:n\in\mathbb{Z}^{d}\right\}}$ forms a countable partition of ${\mathbb{R}^{d}/\mathbb{Z}^{d}}$. For a subset ${E\subset\mathbb{R}^{d}}$, define

$\displaystyle \hat{m}(E):=\sum_{n\in\mathbb{Z}^{d}}\hat{m}_{0}(E\cap Q_{n}-n)$

Observe that if ${E\subset Q_{n}}$, for some ${n\in\mathbb{Z}^{d}}$, then ${\hat{m}(E)=\hat{m}_{0}(E-n)}$. Properties (i) and (ii) are evident from the corresponding properties of ${\hat{m}_{0}}$ and the definition of ${\hat{m}}$.

To prove (iii), we will first show translation invariance for ${h\in\mathbb{Z}^{d}}$. Then by considering ${h\pmod{1}}$, it suffices to show translation invariance for ${h\in Q_{0}}$.
For ${h=m\in\mathbb{Z}^{d}}$, observe that ${(E\cap Q_{n})-n=(E\cap Q_{n-m})-(n-m)}$, whence

$\begin{array}{lcl}\displaystyle \hat{m}(E_{n}+m)&=&\displaystyle\sum_{n\in\mathbb{Z}^{d}}\hat{m}_{0}((E+m)\cap Q_{n}-n)\\[1.5 em]&=&\displaystyle\sum_{n\in\mathbb{Z}^{d}}\hat{m}_{0}(E\cap Q_{n-m}-(n-m))\\[1.5 em]&=&\displaystyle\hat{m}_{0}(E)\end{array}$

since ${n\mapsto n-m}$ is a bijection of ${\mathbb{Z}^{d}}$. Now suppose that ${h=(h_{1},\ldots,h_{d})\in Q_{0}}$. For ${n\in\mathbb{Z}^{d}}$, set ${E_{n}:=E\cap Q_{n}}$ and define sets

$\displaystyle \begin{array}{lcl} \displaystyle E_{n}'&:=&\displaystyle E_{n}\cap (n_{1},n_{1}+1-h_{1}]\times\cdots\times (n_{d},n_{d}+1-h_{d}]\\[1.5 em]\displaystyle E_{n}''&:=&\displaystyle E_{n}\cap (n_{1}+1-h_{1},n_{1}+1]\times\cdots\times (n_{d}+1-h_{d},n_{d}+1]\end{array}$

We can write ${E}$ as the disjoint union ${\bigcup E_{n}'\cup\bigcup E_{n}''}$. Observe that ${(E_{n}+h)\cap Q_{n}\subset Q_{n}}$ while ${E_{n}''+h\subset Q_{\tilde{n}}}$, where ${\tilde{n}=(n_{1}+1,\ldots,n_{d}+1)}$. Using the finite additivity of ${\hat{m}_{0}}$ and the ${\mathbb{Z}^{d}}$-translation invariance of ${\hat{m}}$, we obtain

$\displaystyle \begin{array}{lcl} \displaystyle\hat{m}(E+h)&=&\displaystyle\hat{m}\left(\bigcup_{n\in\mathbb{Z}^{d}}E_{n}'+h\right)+\hat{m}\left(\bigcup_{n\in\mathbb{Z}^{d}}E_{n}''+h\right)\\[2 em]&=&\displaystyle\sum_{n\in\mathbb{Z}^{d}}\hat{m}_{0}((E_{n}'+h)-n)+\sum_{n\in\mathbb{Z}^{d}}\hat{m}_{0}(E_{n}''+h-\tilde{n})\\[2 em]&=&\displaystyle\sum_{n\in\mathbb{Z}^{d}}\left[\hat{m}_{0}(E_{n}'-n)+\hat{m}_{0}(E_{n}''+2h-\tilde{n})\right]\\[2 em]&=&\displaystyle\sum_{n\in\mathbb{Z}^{d}}\left[\hat{m}_{0}(E_{n}'-n)+\hat{m}_{0}(E_{n}''-n)\right]\\[2 em]&=&\displaystyle\sum_{n\in\mathbb{Z}^{d}}\hat{m}_{0}(E_{n}-n)=\hat{m}_{0}(E)\end{array}$

$\Box$

For ${d=1}$, the existence of ${\hat{m}}$ shows that there is no Banach-Tarski paradox in the real line, as the only nontrivial rigid motions are translations. It turns out that the Banach-Tarski paradox is also false in the plane. The interested reader in this result and more may consult Banach’s 1923 article “Sur le probleme de la mesure” and Banach’s 1924 article co-authored with Alfred Tarski. A more recent reference is Stan Wagon’s “The Banach-Tarski Paradox”.

[1] S. Banach, A. Tarski, Sur la decomposition des ensembles de points en parties respectivement congruentes, Fund. math 6 (1924), 118-148.

[2] S. Banach, Sur le probleme de la mesure, Fund. math 4 (1923), 7-33.

[3] E.M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis) (Bk. 4), Princeton UP, 2011.

[4] S. Wagon, The Banach-Tarski Paradox, Cambridge UP, 1993.