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Ladder

Measure — the ladder

6 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

    Almost none of it left, and still uncountably many

    Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

    rung 1 · analysis
  2. A set with no interval in it and half its length left, after 6 stages. Stages of removing a shrinking middle from every surviving interval, with the total length left printed at each stage, and the middle-thirds construction of the same depth drawn beneath for comparison.

    No interval in it, and length to spare

    The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.

    rung 2 · analysis
  3. The rationals covered by intervals of total length 0.1800. Intervals of rapidly shrinking length placed around the rationals of the unit interval in the order they are listed, with the union of them drawn as a single band beneath.

    Covering a set from outside

    To say how long a set is, cover it with intervals and add their lengths, then take the smallest total any covering achieves. That definition is short, obviously right for an interval, and gives the rationals a length of nothing.

    rung 3 · analysis
  4. Two indicators, and the upper sum that will not come down. A partition of the unit interval drawn against the middle-thirds set and against a set of positive length, above a chart of each one's upper sum as the partition is refined.

    Which functions can be added up

    Riemann's integral works when the upper and lower sums close on each other. The exact condition for that, found once measure existed to state it in, is that the points where the function jumps have measure zero — which some nowhere dense sets fail.

    rung 4 · analysis
  5. The classes, a selection from them, and the translates that cannot have a length. Points of several classes of the unit interval under translation by rationals, drawn one class per row, above rows showing rational translates of a selection that never overlap.

    A set that has no size at all

    Slide the unit interval along itself by every rational and the points fall into classes. Choose one point from each and the resulting set has no length — not zero, not positive, none: countably many disjoint copies of it would have total length nought or infinity, and the union needs something in between.

    rung 5 · analysis
  6. A staircase with no steps. The Cantor function drawn to several stages: a continuous non-decreasing curve from nought to one which is constant on every interval of the complement of the middle-thirds set, so its whole rise happens on a set of measure zero.

    A staircase with no steps

    A function that rises from nought to one, is continuous everywhere, and has derivative zero at almost every point. All of its climbing happens on a set of no length at all, which is possible because that set has uncountably many points.

    rung 6 · analysis

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