  Consider the following question: given two functions on a symplectic manifold
whose Poisson bracket is small, is it possible to approximate them in the $C^0$
norm by commuting functions? We give a positive answer in dimension two, as a
particular case of a more general statement which applies to functions on a
manifold with a volume form. This result is based on a lemma in the spirit of
geometric measure theory. We give some immediate applications to function
theory and the theory of quasi-states on surfaces with area forms.
