### On the Baire order of concentrated spaces and ${L}_{1}$ spaces

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We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

ℒ denotes the Lebesgue measurable subsets of ℝ and ${\mathcal{L}}_{0}$ 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$ℒ0$hasaperfectsubsetQ\in \mathcal{L}\${\mathcal{L}}_{0}$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ${\mathcal{L}}_{0}$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $\left({s}^{0}\right)$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

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