# Comparison Of Zermelo's Axiom Of Choice

To illustrate the concept, we can use a physical analogy as follows which was mentioned by Kelly et al., (2012). Assume there

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Then, we can say that both of them are in the same class from the division of real number into an uncountable number of disjoint classes. Any number in [0,1] from each class will form a new set of V⊆[0,1] which consist of the selected numbers (Saucer, 2016)

The connections between those three are

Both constructions of Banach-Tarski Paradox and Vitali sets depend on the axiom of choice in order for them to function well.

Banach-Tarski Paradox and Vitali sets are non-measurable sets and they are two of the consequences of the axiom of choice

We know that the set of real numbers R is uncountable although set is actually a collection of elements. Thus, we cannot measure the size of the size directly. However, we can measure the size using Borel set that we define its measure as a length. From this length, we can simply generalize the set to become Lebesgue measure

Borel measures on R, B(R) is the smallest σ-algebra that contains the open intervals of R as stated in Wikipedia (2016). The Borel measure on real number, R is the choice of Borel measure which assigns μ ((a,b])= b-a for every half-open interval (a,b]. Besides, Weir (1974) mentioned that a Borel measure is any measure μ defined on the σ-algebra of Borel