  In fact, Scott showed that (in contrast to the small large cardinals) measurable cardinals cannot exist in Gödel's constructible universe. Let us be precise about this.  Let V=L be the statement that asserts that all sets are constructible.  Then for each small large cardinal axiom φ (to be precise, those listed above) if the theory ZFC+φ is consistent then so is the theory ZFC+φ+V=L.  In contrast, the theory ZFC + “There is a measurable cardinal” proves ¬V=L.  This may seem somewhat counterintuitive since L contains all of the ordinals and so if κ is a measurable cardinal then κ is an ordinal in L.  The point is that L cannot “recognize” that κ is a measurable cardinal since it is too “thin” to contain the ultrafilter that witnesses the measurability of κ.