For we would like to have a function that assigns to each a number , the n-dimensional measure of , such that is given by the usual integral formulas when the latter apply. The function has the following properties:

1) If is a finite or infinite sequence of disjoint sets, then

.

2) If is congruent to (that is, if can be transformed into by translations, rotations, and reflections), then .

3) , where is the unit cube

for .

However, 1)-3) are mutually inconsistent. To see why, we can define an equivalence relation on by declaring that iff is rational. Let be a subset of that contains precisely one member of each equivalence class. Let , and for each let

To obtain , shift to the right by units and then shift the part that sticks out beyond one unit to the left. Then , every belongs to precisely one . Indeed, if is the element of that belongs to the equivalence class of , then where if or if ; on the other hand, if , then (or ) and (or ) would be distinct elements of belonging to the same equivalence class, which is impossible.

Suppose that now satisfies (i), (ii), and (iii). By (i) and (ii),

for any . Also, since is countable and $[0, 1)$ is the disjoint union of the ‘s,

by (1) again. But by 3), since $\mu(N_r)=\mu(N)$, the sum on the right is either 0 (if ) or (if ). Hence no such can exist.

But in 1924, Banach and Tarski proved the following amazing result:

Let and be arbitrary bounded open sets in , . There exist and subsets of such that:

1) the ‘s are disjoint and their union is ;

2) the ‘s are disjoint and their union is ;

3) is congruent to for .

The construction of sets and depends on the axiom of choice. But their existence precludes the construction of any that assigns positive, finite values to bounded open sets and satisfies (1) for finite sequences as well as (2).

“Don’t take anything Personally Nothing others do is because of you. What others say and do is a projection of their own reality, their own dream. When you are immune to the opinions and actions of others, you won’t be the victim of needless suffering” ~ Miguel Ruiz

To be continued tomorrow 🙂

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 19-21.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.