Tuesday, March 19, 2013

... "i sat in abject horror, my mathematical blood ran cold" ...

I will show you something different from either
        Your shadow at morning striding behind you
                Or your shadow at evening rising to meet you;
                        I will show you fear in a handful of dust. [0]

you know, i had always been dismissive of those constructivists who, among other things, refuse to accept the axiom of choice as part of their proof-writing toolbox: a bunch of mealy-mouthed naysayers and contrarians, i thought!

it had always seemed to me a handy, albeit strange, tool .. but it gets the job done, right?
one of its consequences, the hahη-baηach separation theorem, is incredibly useful .. if not highly magical and never leading to any concrete example. if i can't build something by hand, then usually i use hb.

similarly, for me baηach-alaοglu is like crack: i think i have some mental addiction to weak-star convergent subsequences .. or, if the situation calls for it, nεts. [1]
the last few days have been conceptually difficult .. to the point where i thought i stumbled onto either a paradox or a counterexample to one of my own results.

i felt like my imagination was being stretched to its (very limited) capacity and that i was teetering over the edge of conventional sanity. this afternoon i attained some kind of resolution, though. at the same time,
i sat in abject horror,
my mathematical blood ran cold,
and i was ready to throw down some printed pages to the ground,
step on them repeatedly,
and immediately afterwards, run away, screaming.
fortunately (for my officemate, anyway) and as creepy as this feeling was, i restrained myself and sat calmly at my desk, trying to look at the bright side:

well, at least the theorem's not wrong.

for the record, i've been thinking (too much and too often) about the dual of the βanach space of functions $L^\infty(\mathbb{R}^n)$, which consists of bounded, fιnitely-additive sιgned measures on $\mathbb{R}^n$ that vanish on sets of leþesgue measure zero.

if you have never been curious about these objects, then don't start now. seriously.

although they have a not-unnatural role in functional analysis, my obsession with them has gotten to the point
where i think i have become a generally worse person and perhaps less human.

i learned of the following results from an old paper of hewιtt and yοsida, called finitely additive measures from 1952. in that sense, it reads like a gothic novel .. all quiet and calm at first, and then the monsters come.
theorem 3.3: fubini's theorem fails for finitely additive measures in $[L^\infty(\mathbb{R})]^*$.

theorem 3.4: there exists a nonzero finitely-additive measure $\zeta$ on $\mathbb{R}$ so that $$ \int_{-\infty}^\infty c(t) \, d\zeta(t) \;=\; 0 $$ for all bounded continuous functions $c$ ... and any such measure $\zeta$ must be purely finitely additive.

roughly speaking .. by "purely finitely additive" here, they mean that the object cannot have any nonzero part that behaves like a usual (countably additive) measure. in other words, it's a distinctly exotic object.

theorem 3.6. for any real number $a$, there exists $\zeta_a$ in $[L^\infty(\mathbb{R})]^*$ so that $$ \int_{-\infty}^\infty x(t+u) \, d\zeta_a(u) \;=\; x(t+a) $$ for all essentially bounded functions $x \in L^\infty(\mathbb{R})$ and a.e. $t \in \mathbb{R}$ [2].

keep in mind that $\zeta_a$ is not a point-mass at $a$. in particular, it vanishes on all βorel sets of leþesgue measure zero! (this, by the way, was something close to the paradox i had in mind, believing that it was impossible ..)
thinking about it, the results aren't that much more surprising than the banach-tarsκi paradοx. then again, it's been an obsessive week or two.

i think i need a vacation.

[0] re-reading this line by eliot, notions like "cantοr dust" and singular measures come to mind.
[1] not everything in life is metrisable, you know. for some reason, my work has taken me to these exotic locales, lately.
[2] the original statement was over-simplified. thanks to L for pointing this out.


Daniel said...

I've found the ba space to be fun, but I didn't have to go that far into it. The cases I used also did not have the restriction of absolute continuity with the Lebesgue measure. I'm pretty sure my results hold for ba, but I've just stuck to $ca(\Sigma)$ since it is also weak-* compact. And once I waved the compactness magic wand, I showed that my limit was still in the space I really cared about, L^1.

I would be curious to see your result if/when you get it to a point you are letting it out.

Leonid said...

My reading of Thm 3.6 is different. It asserts equality only for a.e. shift of x (which makes the right-hand side well defined). Also, I don't see the part about vanishing on Borel sets. The proof of Thm 3.6 looks like it produces the point mass at a.

janus said...

sorry for the confusion and misquotes, L., and yes, the shift $t+u$ is missing; i overlooked that part of the proof (in that i was looking too quickly for a $t$ and not a $v$).

as for the "vanishing on Borel sets" of measure zero, i was referring (possibly incorrectly) first to (A) that Lebesgue measure is complete (i.e. all subsets of measure zero are measurable), then to (B) Thm 2.3, where $\mathcal{M}$ is taken to be the $\sigma$-algebra of Borel subsets of $\mathbb{R}$ and $\mathcal{N}$ to be the collection of Borel subsets of Lebesgue measure zero, so $[L^\infty(\mathbb{R})]^*$ would consist of finitely-additive measures that vanish on Borel subsets of Lebesgue measure zero.

Leonid said...

I would say that zeta_a and delta_a are different extensions of the same object. Their common ancestor is the evaluation functional E_a on continuous functions. There are two ways to make this functional act on sets, i.e., make it a measure:

(A) approximate characteristic function of an open set by continuous functions; then approximate measurable sets by open sets. This yields delta_a.

(B) apply Hahn-Banach to extend E_a to L^infty, which includes characteristic functions of measurable sets. This yields zeta_a.

The procedures are different, and so are the results. But zeta_a is the closest thing to delta_a that one can have in (L^infty)*.