I have seen a number of definitions of measurable functions, particularly, for the following two definitions, are they the same? 1) A function f:X->R is measurable iff for every open subset T of R, the inverse image of T is measurable. 2) A function f:X->R is measurable iff for every Borel set B of R, the inverse image of B is measurable. Here, R is the real line. I suppose they are the same, but I haven't been able to show they are equivalent (one direction is easy, but the other one, by assuming any open set has measurable inverse image and showing that the inverse image of all Borel set is measurable, doesn't seem to be that easy, since Borel sets in general cannot be easily writen down in terms of open sets).