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In this paper, we study the supports of measures in the free additive convolution semigroup   $\{\mu ^{\boxplus t}:t&gt;1\}$  , where  μ  is a Borel probability measure on   ${\mathbb R}$  . We give a formula for the density of the absolutely continuous part of   $\mu ^{\boxplus t}$   and use this formula to obtain certain regularizing properties of   $\mu ^{\boxplus t}$  . We show that the number  n ( t ) of the components in the support of   $\mu ^{\boxplus t}$   is a nonincreasing function of  t  and give equivalent conditions so that  n ( t )=1 for sufficiently large  t . Moreover, a measure  μ  so that   $\mu ^{\boxplus t}$   has infinitely many components in the support for all  t &gt;1 is given.

Supports of Measures in a Free Additive Convolution Semigroup