What measure zero means¶

Level: 201 · for anyone who has met a limit

One line: A set has measure zero when it fits inside intervals whose lengths add up to as little as anyone asks — a definition that only ever adds up the lengths of intervals, which is exactly why it works on sets too ragged to measure.

You cannot measure a strange set directly¶

The length of the interval [2, 5] is 3, and nobody argues about that. But what is the length of the set of rational numbers between 0 and 1? That set has no ends and no pieces to lay a ruler along: every stretch of the line, however short, contains some of it and misses some of it. Length in the everyday sense has nothing to grab.

Measure zero sidesteps the question. It never asks for the length of the set. It asks something that can be answered: can the set be hidden inside intervals whose ordinary lengths add up to almost nothing?

What the program prints¶

Verified output of what_measure_zero_means.py — regenerated by tools/run_examples.py, never hand-typed.

1. ONE POINT
   Cover the point 1/2 with an interval as long as the budget allows:
     budget 1/10        interval (9/20, 11/20)                       length 1/10
     budget 1/1000      interval (999/2000, 1001/2000)               length 1/1000
     budget 1/1000000   interval (999999/2000000, 1000001/2000000)   length 1/1000000
   The point's length is at most 1/10, at most 1/1000, at most any
   positive number you care to name. Only one length qualifies: 0.

2. FINITELY MANY POINTS
   Five points, a budget of 1/100. Split it evenly: each point gets 1/500.
     0     inside (-1/1000, 1/1000)
     1/4   inside (249/1000, 251/1000)
     1/2   inside (499/1000, 501/1000)
     3/4   inside (749/1000, 751/1000)
     1     inside (999/1000, 1001/1000)
   total length: 5 x 1/500 = 1/100, within the budget.
   Any finite list of points works the same way: split the budget evenly.

3. THE DEFINITION
   A set has MEASURE ZERO if, for every budget eps > 0, it fits inside
   a list of intervals -- finitely many, or an infinite list -- whose
   lengths add up to at most eps.

   Only the lengths of INTERVALS are ever added. Nobody has to say what
   the 'length' of the set itself is, and that is the whole trick: the
   definition works on sets far too ragged to measure directly.

4. ONE SET OF POINTS, TWO ROOMS
   The points of [0,1], cut into n equal pieces, each piece covered.
   On a line a piece needs an interval. In the plane it needs a square.

             n   each interval   total length   each square       total area
             1   1               1              1                 1
            10   1/10            1              1/100             1/10
           100   1/100           1              1/10000           1/100
         1,000   1/1000          1              1/1000000         1/1000
     1,000,000   1/1000000       1              1/1000000000000   1/1000000

   On the line these covers total 1 for every n. In the plane the total
   shrinks like 1/n, because a square's SECOND side shrinks too:
   n x (1/n)^2 = 1/n.

   So [0,1] has area 0 and length 1. 'Measure zero' always means measure
   zero IN some space: thin compared with the room around it.

5. THE QUESTION THIS LEAVES
   Section 4 had help: a segment is thin in the plane because it has one
   dimension fewer than the plane. Can a set with infinitely many points
   have length zero INSIDE the line itself?
     countably many points     the rationals     (next lesson)
     uncountably many points   the Cantor set    (the one after)

Start with one point¶

Section 1 is the argument this whole chapter runs on, in its simplest form. Name a budget, say 1/10. An open interval of length 1/10 around the point 1/2 covers it. Name a smaller budget — 1/1000, one millionth — and a shorter interval still covers it. So the point's length is at most every positive number anyone can name, and the only length like that is 0.

Five points are no harder (section 2): split the budget evenly. With a budget of 1/100, each point gets an interval of length 1/500, and the five total exactly 1/100. Any finite set of points works the same way.

The definition¶

A set E has measure zero if, for every ε > 0, there is a list of intervals I₁, I₂, I₃, … — finitely many, or infinitely many — that together contain E, with

|I₁| + |I₂| + |I₃| + ⋯ ≤ ε.

Three parts of it are worth reading slowly.

"For every ε." Not "for some small ε". The set has to fit inside total length 1/10, and inside one millionth, and inside any budget at all. A set that fits inside total length 1/10 and no smaller is not thin; it might simply have length 1/10.

"A list." The intervals may be infinitely many, but they must come one after another — a first, a second, a third — so that their lengths can be added as a series. That word does real work, and countable sets shows what goes wrong without it.

"≤ ε." Some books write "< ε" instead. It makes no difference: if every budget can be met, then ε/2 can be met, and ε/2 is less than ε. Whether the intervals are open or closed makes no difference either — widen each one by a tiny amount, a different amount for each, and the total grows by as little as you like.

Notice what the definition never uses: the length of E. Only intervals are ever measured. That is the whole trick, and it is why the definition applies to sets like the ones in the next three lessons, which have no pieces a ruler could be laid along.

One set, two rooms¶

Section 4 takes one set of points, [0, 1], and covers it two ways.

On the line, cut [0, 1] into n pieces and cover each with an interval of length 1/n. The total is n × 1/n = 1, for every n.

In the plane, [0, 1] is a segment lying on the x-axis, and the natural cover is n squares of side 1/n. The total area is n × (1/n)² = 1/n, which goes to 0. A million squares cover the segment with total area one millionth.

So the same points have length 1 and area 0. The squares win because a square has a second side, and the second side shrinks too. This is the precise sense in which a line "has no thickness": it has measure zero in the plane. Measure zero is never a property of a set alone. It belongs to a set inside a space, and the space decides whether intervals, squares or cubes do the covering. With squares and area, or cubes and volume, the same definition is what zero area and zero volume mean.

The table shows only equal-piece covers on the line. That no list of intervals, however clever, can cover [0, 1] with total length less than 1 is a theorem, not something a table can show. Its proof uses the compactness of [0, 1] (the Heine–Borel theorem) to cut any cover down to finitely many intervals, and finitely many intervals covering [0, 1] plainly need total length at least 1. The fat Cantor set lesson leans on this fact.

What this sets up¶

A segment is thin in the plane for an easy reason: it has one dimension fewer than the room around it. The next two lessons ask the hard version of the question. Can a set be thin inside the line itself, where every interval has positive length?

With countably many points, yes — even when they crowd into every stretch of the line: countable sets.
With uncountably many points, still yes: the Cantor set.

Run it yourself¶

From the root of your clone of this repository:

python3 02_Measure_Zero/what_measure_zero_means/examples/what_measure_zero_means.py