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We consider the family 𝒞 ⁢ 𝒮 {\mathcal{CS}} of symmetric Cantor subsets of [ 0 , 1 ] {[0,1]}. Each set in 𝒞 ⁢ 𝒮 {\mathcal{CS}} is uniquely determined by a sequence a = (a n) {a=(a_{n})} belonging to the Polish space X : = (0 , 1) ℕ {X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}} equipped with probability product measure μ. This yields a one-to-one correspondence between sets in 𝒞 ⁢ 𝒮 {\mathcal{CS}} and sequences in X. If 𝒜 ⊂ 𝒞 ⁢ 𝒮 {\mathcal{A}\subset\mathcal{CS}} , the corresponding subset of X is denoted by 𝒜 ∗ {\mathcal{A}^{\ast}}. We study the subfamilies ℋ 0 {\mathcal{H}_{0}} , 𝒮 ⁢ 𝒫 {\mathcal{SP}} and ℳ {\mathcal{M}} of 𝒞 ⁢ 𝒮 {\mathcal{CS}} , consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have ℳ ⊂ ℋ 0 ⊂ 𝒮 ⁢ 𝒫 {\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}} , and these inclusions are proper. We prove that the sets ℳ ∗ {\mathcal{M}^{\ast}} , ℋ 0 ∗ {\mathcal{H}_{0}^{\ast}} , 𝒮 ⁢ 𝒫 ∗ {\mathcal{SP}^{\ast}} are residual in X, and μ ⁢ (ℋ 0 ∗) = 0 {\mu(\mathcal{H}_{0}^{\ast})=0} , μ ⁢ (𝒮 ⁢ 𝒫 ∗) = 1 {\mu(\mathcal{SP}^{\ast})=1}. [ABSTRACT FROM AUTHOR]

It is shown that well-orderings are either negligible or nonmeasurable with respect to the completions of σ-finite product measures. Several consequences of this fact are discussed in the light of some classical problems formulated by Hilbert, Lebesgue and Luzin. [ABSTRACT FROM AUTHOR]