Posts tagged with measure theory

population map of Scotland

population map of Scotland

(Source: Wikipedia)


hi-res




Lebesgue’s approach to integration was summarized in a letter to Paul Montel. He writes:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
Siegmund-Schultze, Reinhard (2008), “Henri Lebesgue”, in Timothy Gowers, June Barrow-Green, Imre Leader, Princeton Companion to Mathematics

(Source: Wikipedia)




One

Counting generates from the programmer’s successor function ++ and the number one. (You might argue that to get out to infinity requires also repetition. Well every category comes with composition by default, which includes composition of ƒ∘ƒ∘ƒ∘….)

But getting to one is nontrivial. Besides the mystical implications of 1, it’s not always easy to draw a boundary around “one thing”. Looking at snow (without the advantage of modern optical science) I couldn’t find “one snow”. Even where it is cut off by a plowed street it’s still from the same snowfall.
a larger &lsquot;thing&rsquot; with holes in it ... like the snow has &lsquot;road holes&rsquot; in it
And if you got around on skis a lot of your life you wouldn’t care about one snow-flake (a reductive way to define “one” snow), at least not for transport, because one flake amounts to zero ability to travel anywhere. Could we talk about one inch of snow? One hour of snow? One night of snow?

http://2.bp.blogspot.com/_sf5B7n5Avcg/THFKAnY0TCI/AAAAAAAAAmI/x9E4slMs0uQ/s1600/IMGP1553.JPG

Speaking of the cold, how about temperature? It has no inherent units; all of our human scales pick endpoints and define a continuum in between. That’s the same as in measure theory which gave (along with martingales) at least an illusion of technical respectability to the science of chances. If you use Kolmogorov’s axioms then the difficult (impossible?) questions—what the “likelihood” of a one-shot event (like a US presidential election) actually means or how you could measure it—can be swept under the rug whilst one computes random walks on trees or Gaussian copulæ. Meanwhile the sum-total of everything that could possibly happen Ω is called 1.

With water or other liquids as well. Or gases. You can have one grain of powder or grain (granular solids can flow like a fluid) but you can’t have one gas or one water. (Well, again you can with modern science—but with even more moderner science you can’t, because you just find a QCD dynamical field balancing (see video) and anyway none of the “one” things are strictly local.)

And in my more favourite realm, the realm of ideas. I have a really hard time figuring out where I can break off one idea for a blogpost. These paragraphs were a stalactite growth off a blobular self-rant that keeps jackhammering away inside my head on the topic of mathematical modelling and equivalence classes. I’ve been trying to write something called “To equivalence class” and I’ve also been trying to write something called “Statistics for People Who Program Computers” and as I was talking this out to myself, another rant squeezed out between my fingers and I knew if I dropped the other two I could pull One off it could be sculpted into a readable microtract. Leaving “To Equivalence Class”, like so many of the harder-to-write things, in the refrigerator—to marinate or to mould, I don’t know which.

But notice that I couldn’t fully disconnect this one from other shared-or-not-shared referents. (Shared being English language and maybe a lot of unspoken assumptions we both hold. Unshared being my own personal jargon—some of which I’ve tried to share in this space—and rants that continually obsess me such as the fallaciousness of probabilistic statements and of certain economic debates.) This is why I like writing on the Web: I can plug in a picture from Wikipedia or point back to somewhere else I’ve talked on the other tangent so I don’t ride off on the connecting track and end up away from where I tried to head.

The difficulty of drawing a firm boundary of "one" to begin the process of counting may be an inverse of the "full" paradox or it may be that certain things (like liquid) don’t lend themselves to counting in an obvious way—in jargon, they don’t map nicely onto the natural numbers (the simplest kind of number). If that’s a motivation to move from discrete things to continuous when necessary, then I feel a similar motivation to move from Euclidean to Hausdorff, or from line to poset. Not that the simpler things don’t deserve as well a place at the table.

We thinkers are fairly free to look at things in different ways—to quotient and equivalence-class creatively or at varying scales. And that’s also a truth of mathematical modelling. Even if maths seems one-right-answer from the classroom, the same piece of reality can bear multiple models—some refining each other, some partially overlapping, some mutually disjoint.




In the Public Encyclopedia’s (present) discussion of the hypothetical existence of a magnetic monopole

http://upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Em_monopoles.svg/1000px-Em_monopoles.svg.png

in nature, among the possible fundamental particles, exemplifies both (and maybe >2) “sides” in the debate over what probability means:

Magnetism in bar magnets and electromagnets does not arise from magnetic monopoles, and in fact there is no conclusive experimental evidence that magnetic monopoles exist at all in the universe.

Since Dirac’s 1931 paper[8] , several systematic monopole searches have been performed. Experiments in 1975[10] and 1982[11] produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.[12]Therefore, it remains an open question whether or not monopoles exist.

Further advances in theoretical particle physics, particularly developments in grand unified theories and quantum gravity, have led to more compelling arguments[which?] that monopoles do exist. Joseph Polchinski, a string-theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen”.[13]These theories are not necessarily inconsistent with the experimental evidence. In some theoretical models, magnetic monopoles are unlikely to be observed, because they are too massive[why?] to be created in particle accelerators, and also too rare in the Universe to enter a particle detector with much probability.[13] (According to these models, there may be as few as one monopole in the entire visible universe.[14])

http://upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Em_dipoles.svg/1000px-Em_dipoles.svg.png

Here are a few potential explanations of how one is to arrive at a probability number:

  • opinion — it’s just Joseph Polchinski’s opinion
  • frequentism — Europeans never observed a black swan before exploring the New World, therefore black swans have 0% chance of existing.
  • frequentism + how hard you’ve searched — the probability comes attached with a confidence number. If you’ve stayed within the city limits of Minneapolis your entire life, you should attach a low confidence to your search for tarantulas the size of your head. But we’ve tried very hard to find monopoles, and haven’t. So a “more confident” zero on that one.
  • Dutch Books — could we arbitrage Joseph Polchinski’s “sure thing” bet?
  • authority, credibility, expertise — who exactly is this Joseph Polchinski character, anyway? And who says he’s such an expert? Is he an interested party? I don’t believe what vested interests and biased sources say, even if it happens to be true.
  • propensity — good gravy, I don’t even get to invoke the famous “coin has an innate propensity to tend to certain heads/tails ratio” because it would get us nowhere in terms of “Do monopoles have a propensity to exist or not?”. Anyway propensity merely passes the buck even in the cases where it does make sense.
  • reason & facts — there is no conclusive evidence that monopoles exist, yet they haven’t been proven impossible. I will withhold my opinion and it would be unreasonable to assign a probability mass to either alternative. We’re simply somewhere ∈ [0%, 100%] at this time.
  • model strength — some of these models sound suspect. It’s constructed “just so” that there’s only one monopole in the universe? Very convenient for you, when you want to say monopoles exist and we just haven’t seen them yet. Pull the other one!

image
All of the stochastic maths is done with the Kolmogorov axioms, i.e. it’s done with measure spaces with a fixed | finite | constant measure (= 100% of the probability mass) without connecting that to “how likely” a one-off event is. (Much like some maths you could pass off as financial modelling “is just" the theory of martingales = fair repeated bets.) But it needn’t have be called “likelihood”, it could have been “fuzzy truthiness” or “believability” or “motions of a fixed-volume-but-infinitely-divisible liquid”. As Cosma Shalizi puts it here:

Probabilities are numbers that tell us how often things happen.

Mathematicians are anxious to get on with talking about ergodicity, Markov transition matrices, and large-deviations theory. What you’re seeing in this block quote is the handoff between mathematicians and philosophers—essentially the mathmos say “You take it from here to the firm foundation” and philosophers, so far, haven’t been able to.

 

Is there a problem in practice due to not having a sound foundation on our concept of probability? Yes. It’s not secure to move forward with the rear flank uncovered. The lax attitude toward probability and “We’ll do the best with what we can” lets us make up numbers for the {pessimistic, neutral, optimistic} scenarios of our forecasting spreadsheets.

Think about when some consequential decision by a powerful group depends on the value of one parameter. It could be

  • the likelihood of Floridian home prices decreasing by more than 5% in a year,
    image
  • the likelihood of [foreign country X] attacking "us" in response to Y,
  • the likelihood of RHIC creating a strangelet and swallowing the world in a minisecond,
  • the likelihood of construction on the new power plant going over budget,
  • the likelihood of borrowing rates staying this low for another 5 years,
    image
  • the likelihood of real GDP rising at least 2%/year during the next 10 years,
  • the likelihood of our borrowing rate quadrupling
    image
  • the likelihood that your college degree will “be worth it” to you
  • the likelihood of this whole startup thing actually working.
    http://paulgstacey.files.wordpress.com/2010/09/startup_financing_cycle.png

and I get to either rely on

  • historical data (“home prices have always gone up before”, “we haven’t seen any problems with financial derivatives yet”, “correlation with a Gaussian copula has always worked so far”),
  • reason and facts (and multiply an endless debate among the experts),
  • or gut (throw in some numbers that sound pessimistic, optimistic, and neutral, and we’ll see how the forecast behaves).

We got nuthin’.




For those not in the know, here’s what mathematicians mean by the word “measurable”:

  1. The problem of measure is to assign a ℝ size ≥ 0 to a set. (The points not necessarily contiguous.) In other words, to answer the question:
    How big is that?
  2. Why is this hard? Well just think about the problem of sizing up a contiguous ℝ subinterval between 0 and 1.
    image
    • It’s obvious that [.4, .6] is .2 long and that
    • [0, .8] has a length of .8.
    • I don’t know what the length of √2√π/3] is but … it should be easy enough to figure out.
    • But real numbers can go on forever: .2816209287162381682365...1828361...1984...77280278254....
    • Most of them (the transcendentals) we don’t even have words or notation for.
      most of the numbers are black = transcendental
    • So there are a potentially infinite number of digits in each of these real numbers — which is essentially why the real numbers are so f#cked up — and therefore ∃ an infinitely infinite number of numbers just between 0% and 100%.


    Yeah, I said infinitely infinite, and I meant that. More real numbers exist in-between .999999999999999999999999 and 1 than there are atoms in the universe. There are more real numbers just in that teensy sub-interval than there are integers (and there are integers).

    In other words, if you filled a set with all of the things between .99999999999999999999 and 1, there would be infinity things inside. And not a nice, tame infinity either. This infinity is an infinity that just snorted a football helmet filled with coke, punched a stripper, and is now running around in the streets wearing her golden sparkly thong and brandishing a chainsaw:
    I think the analogy of 5_1 to Patrick Bateman is a solid and indisputable one.


    Talking still of that particular infinity: in a set-theoretic continuum sense, ∃ infinite number of points between Barcelona and Vladivostok, but also an infinite number of points between my toe and my nose. Well, now the simple and obvious has become not very clear at all!
    Europe  Data set:> eurodist                 Athens Barcelona Brussels Calais Cherbourg Cologne CopenhagenBarcelona         3313                                                       Brussels          2963      1318                                             Calais            3175      1326      204                                    Cherbourg         3339      1294      583    460                             Cologne           2762      1498      206    409       785                   Copenhagen        3276      2218      966   1136      1545     760           Geneva            2610       803      677    747       853    1662       1418Gibraltar         4485      1172     2256   2224      2047    2436       3196Hamburg           2977      2018      597    714      1115     460        460Hook of Holland   3030      1490      172    330       731     269        269Lisbon            4532      1305     2084   2052      1827    2290       2971Lyons             2753       645      690    739       789     714       1458Madrid            3949       636     1558   1550      1347    1764       2498Marseilles        2865       521     1011   1059      1101    1035       1778Milan             2282      1014      925   1077      1209     911       1537Munich            2179      1365      747    977      1160     583       1104Paris             3000      1033      285    280       340     465       1176Rome               817      1460     1511   1662      1794    1497       2050Stockholm         3927      2868     1616   1786      2196    1403        650Vienna            1991      1802     1175   1381      1588     937       1455                Geneva Gibraltar Hamburg Hook of Holland Lisbon Lyons MadridBarcelona                                                                   Brussels                                                                    Calais                                                                      Cherbourg                                                                   Cologne                                                                     Copenhagen                                                                  Geneva                                                                      Gibraltar         1975                                                      Hamburg           1118      2897                                            Hook of Holland    895      2428     550                                    Lisbon            1936       676    2671            2280                    Lyons              158      1817    1159             863   1178             Madrid            1439       698    2198            1730    668  1281       Marseilles         425      1693    1479            1183   1762   320   1157Milan              328      2185    1238            1098   2250   328   1724Munich             591      2565     805             851   2507   724   2010Paris              513      1971     877             457   1799   471   1273Rome               995      2631    1751            1683   2700  1048   2097Stockholm         2068      3886     949            1500   3231  2108   3188Vienna            1019      2974    1155            1205   2937  1157   2409                Marseilles Milan Munich Paris Rome StockholmBarcelona                                                   Brussels                                                    Calais                                                      Cherbourg                                                   Cologne                                                     Copenhagen                                                  Geneva                                                      Gibraltar                                                   Hamburg                                                     Hook of Holland                                             Lisbon                                                      Lyons                                                       Madrid                                                      Marseilles                                                  Milan                  618                                  Munich                1109   331                            Paris                  792   856    821                     Rome                  1011   586    946  1476               Stockholm             2428  2187   1754  1827 2707          Vienna                1363   898    428  1249 1209      2105  Multi-dimensional scaling of the distances:  > cmdscale(eurodist)                        [,1]        [,2]Athens           2290.274680  1798.80293Barcelona        -825.382790   546.81148Brussels           59.183341  -367.08135Calais            -82.845973  -429.91466Cherbourg        -352.499435  -290.90843Cologne           293.689633  -405.31194Copenhagen        681.931545 -1108.64478Geneva             -9.423364   240.40600Gibraltar       -2048.449113   642.45854Hamburg           561.108970  -773.36929Hook of Holland   164.921799  -549.36704Lisbon          -1935.040811    49.12514Lyons            -226.423236   187.08779Madrid          -1423.353697   305.87513Marseilles       -299.498710   388.80726Milan             260.878046   416.67381Munich            587.675679    81.18224Paris            -156.836257  -211.13911Rome              709.413282  1109.36665Stockholm         839.445911 -1836.79055Vienna            911.230500   205.93020  Plot       require(stats)     loc <- cmdscale(eurodist)     rx <- range(x <- loc[,1])     ry <- range(y <- -loc[,2])     plot(x, y, type="n", asp=1, xlab="", ylab="")     abline(h = pretty(rx, 10), v = pretty(ry, 10), col = "light gray")     text(x, y, labels(eurodist), cex=0.8)
    So it’s a problem of infinities, a problem of sets, and a problem of the continuum being such an infernal taskmaster that it took until the 20th century for mathematicians to whip-crack the real numbers into shape.
  3. If you can define “size” on the [0,1] interval, you can define it on the [−535,19^19] interval as well, by extension.

    If you can’t even define “size” on the [0,1] interval — how do you think you’re going to define it on all of ℝ? Punk.
  4. A reasonable definition of “size” (measure) should work for non-contiguous subsets of ℝ such as “just the rational numbers” or “all solutions to cos² x = 0(they’re not next to each other) as well.

    Just another problem to add to the heap.
  5. Nevertheless, the monstrosity has more-or-less been tamed. Epsilons, deltas, open sets, Dedekind cuts, Cauchy sequences, well-orderings, and metric spaces had to be invented in order to bazooka the beast into submission, but mostly-satisfactory answers have now been obtained.

    It just takes a sequence of 4-5 university-level maths classes to get to those mostly-satisfactory answers.
    image
    One is reminded of the hypermathematicians from The Hitchhiker’s Guide to the Galaxy who time-warp themselves through several lives of study before they begin their real work.

imageimage

For a readable summary of the reasoning & results of Henri Lebesgue's measure theory, I recommend this 4-page PDF by G.H. Meisters. (NB: His weird ∁ symbol means complement.)

That doesn’t cover the measurement of probability spaces, functional spaces, or even more abstract spaces. But I don’t have an equally great reference for those.

Oh, I forgot to say: why does anyone care about measurability? Measure theory is just a highly technical prerequisite to true understanding of a lot of cool subjects — like complexity, signal processing, functional analysis, Wiener processes, dynamical systems, Sobolev spaces, and other interesting and relevant such stuff.

It’s hard to do very much mathematics with those sorts of things if you can’t even say how big they are.




Are mathematicians deliberately obscure? Or is it really so hard for them to write prose?

Check out this description of σ-algebras from Wik***dia.
BAD:

In mathematics, a σ-algebra (also sigma-algebraσ-fieldsigma-field) is a technical concept for a collection of sets satisfying certain properties.[1] The main use of σ-algebras is in the definition of measures; specifically, a σ-algebra is the collection of sets over which a measure is defined.

No sh_t? It’s technical? And it satisfies properties. You don’t say.

The kernel of that paragraph is just one sentence.
GOOD:

In mathematics, a σ-algebra is a measurable collection of sets.

I changed the W*****dia page at 8:40pm on 3 Mar ‘11. Let’s see if I get in trouble. (I bet if I do it will be for “not being rigorous” or “original research”.)

Yes this is a specialist topic, but that doesn’t require gobbledegook. A σ-algebra is measurable like , but is not ℝ. Why can’t we just use normal words?



UPDATE: It hurts to be this right. My changes were reverted about an hour after I put them up. Am I wrong here?

I’m reminded of a story Doug Hofstadter told us about a friend of his who submitted an article in clear, everyday language to an academic journal. According to DH, the journal’s editors rejected the piece, saying it was too unprofessional. They confused jargon with sophistication, bombast with wisdom.

I don’t know the friend’s name or the journal’s name, and I half-wonder if I am just being a pr$ck about this Wikipedia article. But no, think about how people react to the word “maths”. This has got to be the reason—this and boring maths classes. Mathematicians literally refuse to write simply.

UPDATE 2: Another offender is the article on compact topological spaces. I’m actually removing some text from the garbled lede when I say:

In mathematics, specifically general topology, a compact topology is a topological space whose topology has the compactness property.

I think I’ve found a new candidate for worst sentences in the English language. Does anyone have George Orwell’s e-mail address?