I'm trying to directly show that non-computable c.e. sets are Kurtz random, without using the concept of genericity, but to little success. I assume by way of contradiction that $\emptyset'$ (for example) is not included in some c.e. class $W \subset 2^\omega$ whose measure is 1. However, I have no idea how I can exploit the condition about the measure.
Is it easy to see the proposition directly? If so, I'd be grateful if you can provide me with a clue. (If not, I'm happy with the proof via genericity.)
No c.e. set is Kurtz random because any infinite c.e. set has an infinite computable subset and then a test consisting of clopen sets can easily be built to zoom in on that infinite computable subset. I'll provide more details if you'd like.