There is a general connection between notions of forcing adding real numbers and notions of measurability on the real line. Using general results by Ikegami on the relationship between the measurability of Delta^1_2 and Sigma^1_2 sets and the existence of quasi-generics over models of the type L[x], we characterize the statements "every Delta^1_2 set has the Baire property in the eventually different topology" and "every Sigma^1_2 set has the Baire property in the eventually different topology". This is joint work with Jörg Brendle (Kobe).