The Fourier–Faber–Schauder Series Unconditionally Divergent in Measure

详细

We prove that, for every ε ∈ (0, 1), there is a measurable set E ⊂ [0, 1] whose measure |E| satisfies the estimate |E| > 1−ε and, for every function f ∈ C_[0,1], there is ˜ f ∈ C_[0,1] coinciding with f on E whose expansion in the Faber–Schauder system diverges in measure after a rearrangement.