The Lusin Separation Theorem is one of the fundamental early results of classical descriptive set theory. It states that if A1, A2 are disjoint analytic subsets of Baire space then they are Borel separable. Yiannis Moschovakis gives two proofs in his book, “Descriptive Set Theory”. One proof is obviously highly non-constructive. The other would appear to be constructive and uses Bar Induction a...

A theorem of Lusin is proved in the non-ordered context of JB∗-triples. This is applied to obtain versions of a general transitivity theorem and to deduce refinements of facial structure in closed unit ballls of JB∗-triples and duals.

The Continuum Hypothesis implies an Erdös-Sierpiński like duality between the ideal of first category subsets of R , and the ideal of countable dimensional subsets of R. The algebraic sum of a Hurewicz subset a dimension theoretic analogue of Sierpinski sets and Lusin sets of R with any compactly countable dimensional subset of R has first category.

We consider a Cauchy problem for a fractional semilinear differential inclusions involving Caputo’s fractional derivative in non separable Banach spaces under Filippov type assumptions and we prove the existence of solutions. MSC: 34A60, 26A33, 34B15 keywords: fractional derivative, fractional semilinear differential inclusion, Lusin measurable multifunctions.

This survey article gives a history of the development of the theory con cerning restriction theorems in real analysis The discussion will span the history from Lusin s Theorem through very recent results concerning in tersections of Lipschitz smooth and H older class functions