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Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

Jack Brown; Hussain Elalaoui-Talibi — 1999

Colloquium Mathematicae

ℒ denotes the Lebesgue measurable subsets of ℝ and $ℒ_{0}$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 $h a s a p e r f e c t s u b s e t Q \in ℒ $ ℒ_{0}$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes $ℒ_{0}$ ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $(s^{0})$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

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