  The equivalence of the characteristic function approach and the probabilistic
approach to monotone and boolean convolutions is proven for non-compactly
supported probability measures. A probabilistically motivated definition of the
multiplicative boolean convolution of probability measures on the positive
half-line is proposed. Unlike Bercovici's multiplicative boolean convolution it
is always defined, but it turns out to be neither commutative nor associative.
Finally some relations between free, monotone, and boolean convolutions are
discussed.
