This should really be well-known, but I was not able to find a definite answer to this question:

*Is the Fourier transform of a bounded function always a borel measure (i.e. an order 0 distribution)?*

In some sense, the distributional order corresponds to the order of a bounding polynomial and a bounded function can be bounded by an order zero polynomial. But I could not find any reference.

If the above statement is false, what is a counterexample?

If the statement is true: How does bounded variation correspond to all this? I.e. are there criterions for the Fourier transform to be of bounded variation?