Ok, what I mean was f doesn't include b. Because b is a vector field. And grad only maps scalar functions to vectors, but it doesn't map vector functions to anything. Or do you mean E[grad(b_i).b]=0? In that case I don't know why it follows b=0.
The fact that E[b(X)] = 0 is easy, just take f=x_i for each i it is ok. However, I am not sure how to prove Var[b(X)] = 0.
I am not sure what is meant by grad(b).b=0 since b is a vector field, and grad is only defined for scalar functions.
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 R^n. I have also found that for all continuous differentiable function f mapping X to R, the...