In Oxtoby's "Measure and [Baire] Category" it is mentioned that facts such as "Elements with property X form a set of measure zero/first category/countable" can be seen as existence results, as in the proof that there are many irrational numbers, since the set of rational numbers is "small" in all three senses, while the set of real numbers is "big".
From that point of view this three "complicated" ideas, [the Baire Category theorem, naive set theory and Lebesgue measure] introduce very simple ways of proving the existence of potentially complicated objects.