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.

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