A projection of a measurable set is measurable. Not only students believe this. I was asked once (the quote is not precise): "Why do you need this assumption of a measurability of projection? It follows from ..."

Another one seems to be more specific, I just recalled it reading [this][1] example. A sub-$\sigma$-algebra of a countably generated $\sigma$-algebra is countably generated. 


  [1]: http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23493#23493