In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...) prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$ . In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property " $F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$ . In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$. (shrink)

Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: , and.

We show the consistency of ${\frak o} <{\frak d}$ where ${\frak o}$ is the size of the smallest off-branch family, and ${\frak d}$ is as usual the dominating number. We also prove the consistency of ${\frak b} < {\frak a}$ with large continuum. Here, ${\frak b}$ is the unbounding number, and ${\frak a}$ is the almost disjointness number.

We show that is consistent with the existence of a -definable wellorder of the reals and a -definable ω-mad subfamily of [ω]ω.

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω together with.

We prove that the cofinality of the smallest covering of R by meager sets is bigger than the additivity of measure.

We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on ω2.