From the authors’ abstract: We extend von Neumann’s theory of spectral sets, in order to deal with the numerical range of operators. An integral representation for arbitrary operators is given, allowing to extend functional calculus to non-normal operators. We apply our results to the proof of the Burkholder conjecture: let \(T\) be an operator consisting in a finite product of conditional expectation, then for any square integrable function \(f\), the iterates \(T^nf\) converge almost surely to some limit.