I am doing my research in probability. I have found some probability distribution of a random variable X on the n dimensional unit sphere. Let b be a smooth and lipschitz vector field mapping X to [itex]R^n[/itex]. I have also found that for all continuous differentiable function f mapping X to [itex]R[/itex], the expectation of [itex]\triangledown f\cdot b[/itex] is zero. I have strong feeling that this implies b(X)=0 with probability 1, but I am not sure how I can prove it.