This thread is more about language clarification. In one paper it is stated that "to the continuity points of the distribution of $z = a \cdot x $ induced by the measure $ \mu $ on $R^d $ ". No further information about $ \mu $. a is a scalar and z, x are random variables.
I think it means if we have a the Function $ f(b) = P ( [ - \infty ; b] ) = \int_{ - \infty } ^b a \cdot x \, d\mu (dx) $ then we look at the b where f is continue.