Why sigma fields for lebesgue integration?

Seriously?  What's wrong with power sets?  I've never had anyone actually explain to me why we can't define a measure on a power set and have to fuck around with sigma fields instead.

Uh, who told you you can't define a measure on a power set? Power sets are trivially sigma algebras.
However, the Lebesgue measure does not work for all sets in the reals (in ZFC), so you cannot define the Lebesgue measure on the power set of the reals.