Matpic
mathematics, in pictures
Ladder

Measure — the ladder

One essay so far against this idea. A ladder is the distinct arguments that stand against one idea, and this one has room to grow.
  1. the interval1.0000 leftstage 10.6667 leftstage 20.4444 leftstage 30.2963 leftstage 40.1975 leftstage 50.1317 leftstage 60.0878 leftat stage 6 the 64 surviving intervals have total length 0.08779, and 0.91221 has been taken outthe removed pieces are a geometric series — a third, then two ninths, then four twenty-sevenths — which adds to exactly one,so what is left has no length at all

    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

All ladders