  This article presents extensions of the Cram{\'e}r-Wold theorem to measures
that may have infinite mass near the origin. Corresponding results for
sequences of measures are presented together with examples showing that the
assumptions imposed are sharp. The extensions build on a number of results and
methods concerned with injectivity properties of the Radon transform. Using a
few tools from distribution theory and Fourier analysis we show that the
presented injectivity results for the Radon transform lead to Cram{\'e}r-Wold
type results for measures. One purpose of this article is to contribute to
making known to probabilists interesting results for the Radon transform that
have been developed essentially during the 1980ies and 1990ies.
