There exist sets which are not Lebesgue
Define an equivalence relation ~ on
This relation divides into
equivalence classes . Clearly each equivalence
class contains a point in the interval .
Let be a set which contains exactly one
point from each equivalence class. (By strong axiom of choice
For all , define
We observe that
The second inclusion is clear. To see the first we
note that if , then there must be a
must belong some equivalence class and by our definition contains
one element from every equivalence class). Then
for some and thus we have
We note also that
Otherwise for some we would
have , which is impossible by our choice
Now suppose is measurable.
Then every are.
Using (2) and monotonicity property of the Lebesgue
measure we would have
Also by translation invariance of the Lebesgue measure
Therefore, using the fact that
are pairwise disjoint and that the Lebesgue measure is -additive,
we get that
which contradicts (4).
Thus cannot be measurable.