Nikodym Set

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Otto Nikodym (1927) mengkonstruksi himpunan N ⊂ [0,1]² dengan Lebesgue measure = 1 (hampir seluruh persegi), namun untuk setiap titik p ∈ N, ada garis yang melewati p sedemikian rupa sehingga hampir seluruh garis berada di luar N.

Ini tampak mustahil secara intuitif: himpunan yang "mengisi" hampir seluruh persegi namun setiap titiknya memiliki garis hampir-seluruhnya di luar himpunan. Konstruksi ini mengekspos sifat aneh himpunan terukur dengan measure penuh — mereka bisa memiliki struktur geometris yang sangat berlubang-lubang. Relevan dalam analisis harmonik dan teori diferensiasi.

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