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In 1917, Lusin and Sierpinski showed that the unit interval $[0,1]$ can be partitioned into $2^{\aleph_{0}}$ many pairwise disjoint sets each having Lebesgue outer measure 1; say, $X_{i}$, $i \in I$. Fix $i_{0} \in I$. For each proper subset $J$ with $i_{0} \in notin J \subset I$, let $S_{J} = \bigcup_{j \in J} X_{j}$. Then the sets $S_{J}$, $i_{0} \in notin J \subset I$, are distinct modulo $\mathcal{M}$.

Nikolai N. Lusin and Waclaw Sierpinski, “Sur une décomposition d’un intervalle en une infinité non dénombrable d’ensembles non mesurables” [On a decomposition of an interval into a nondenumerably many nonmeasurable sets], Comptes Rendus Académie des Sciences (Paris) 165 (1917), 422-424.

3 added 24 characters in body

In 1917, Lusin and Sierpinski showed that the unit interval $[0,1]$ can be partitioned into $2^{\aleph_{0}}$ many pairwise disjoint sets each having Lebesgue outer measure 1; say, $X_{i}$, $i \in I$. Fix $i_{0} \in I$. For each nonempty proper subset $J$ with $i_{0} \in J \subset I$, let $S_{J} = \bigcup_{j \in J} X_{j}$. Then the sets $S_{J}$, $\emptyset i_{0} \neq in J \subset I$, are distinct modulo $\mathcal{M}$.

Nikolai N. Lusin and Waclaw Sierpinski, “Sur une décomposition d’un intervalle en une infinité non dénombrable d’ensembles non mesurables” [On a decomposition of an interval into a nondenumerably many nonmeasurable sets], Comptes Rendus Académie des Sciences (Paris) 165 (1917), 422-424.

In 1917, Lusin and Sierpinski showed that the unit interval $[0,1]$ can be partitioned into $2^{\aleph_{0}}$ many pairwise disjoint sets each having Lebesgue outer measure 1; say, $X_{i}$, $i \in I$. For each nonempty proper subset $J \subseteq subset I$, let $S_{J} = \bigcup_{j \in J} X_{j}$. Then the sets $S_{J}$, $\emptyset \neq J \subseteq subset I$, are distinct modulo $\mathcal{M}$.