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.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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\$ℒ_0$ has a perfect subset Q ∈ 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 a perfect subset Q which is a subset of or misses M. M ∈ $(s^0)$ if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of $G_δ$ sets which can be used to give similar "Marczewski-Burstin-like" characterizations of the collections $B_w$ (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of $F_σ$ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of $B_r$ (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and $U_0$ (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW