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A non-measurable set in (0,1]

Let "+" stand for addition modulo 1 in (0, 1]. For example, 0.5 + 0.7 = 0.2,
instead of 1.2. If A ⊆ (0, 1], and x is a number , then A+x stands for the set of
all numbers of the form y + x where y ∈ A. You may want to visualize (0, 1] as
a circle that wraps around so that after 1, one starts again at 0.

Define x and y to be equivalent if x +r = y for some rational number r. Then,
(0, 1] can be partitioned into equivalent classes . (That is, all elements in the
same equivalence class are equivalent , elements belonging to different equivalent
classes are not equivalent , and every x ∈ (0, 1] belongs to some equivalence
class.) Let us pick exactly one element from each equivalence class, and let
H be the set of the elements picked this way. (This fact that a set H can be
legitimately formed this way involves the Axiom of Choice, a generally accepted
axiom of set theory.) We will now consider the sets of the form H + r, where r
ranges over the rational numbers in (0, 1]. Note that there are countably many
such sets.

The sets H + r are disjoint. (Indeed, if r1 ≠ r2 and H + r1 and H + r2
share the point h1 + r = h2 + r2, then h1 and h2 differ by a rational number
and therefore are equivalent . If h1 ≠ h2, this contradicts the construction of H,
which contains only one element from each equivalence class. If h1 = h2, then
r1 = r2, which is again a contradiction.) Therefore, (0, 1] is the union of the
countably many disjoint sets H + r.

The sets H + r for different r are "translations" of each other (they are all
formed by starting from the set H and adding a number . The "uniform" proba-
bility measure (or Lebesgue measure) assigns a probability to each interval equal
to its length, so that when an interval is a translation of another, they should
have the same probability. We are interested in whether Lebesgue measure can
be defined for all subsets of (0, 1], while remaining translation-invariant. If this
were possible, each set H +r should have the same probability, and their prob-
abilities should add to 1. But this is impossible, since there are infinitely many
such sets.

A stronger statement is actually true, but harder to prove. there exists no
probability measure on under which P({x}) = 0 for all points x.

The Banach-Tarski Paradox. Let S be the two-dimensional surface of
the unit sphere in three dimensions. There exists a subset F of S such that for
any k ≥ 3,

where each is a rigid rotation. For example, S can be made up by three rotated
copies of F (suggesting probability equal to 1/3, but also by four rotated copies
of F, suggesting probability equal to 1/4). Ordinary geometric intuition clearly
fails when dealing with arbitrary sets.

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