  A real number $x$ is normal with respect to an integer base $b \geq 2$ if its
digit expansion in this base is ``equitable'', in the sense that for $k \geq
1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in
the digit expansion of $x$ with the same limiting frequency. Borel's classical
result \cite{b09} asserts that Lebesgue-almost every number $x$ is normal in
every base $b \geq 2$. This three-part article considers sets of partial
normality. Given any choice of integer bases $\mathscr{B}, \mathscr{B}'
\subseteq \{2, 3, \ldots\}$, we investigate measure-theoretic properties of the
set $\mathscr{N}(\mathscr{B}, \mathscr{B}')$, whose members are, by definition,
normal in the bases of $\mathscr{B}$ and non-normal in the bases of
$\mathscr{B}'$. A pair of sets $(\mathscr{B}, \mathscr{B}')$ is compatible if
any $(b, b') \in \mathscr{B} \times \mathscr{B}'$ is multiplicatively
independent. For compatible $(\mathscr{B}, \mathscr{B}')$ with $\mathscr{B}'
\ne \emptyset$, we construct singular probability measures supported on
$\mathscr N(\mathscr{B}, \mathscr{B}')$ that are both Frostman and Rajchman,
extending prior work of Pollington \cite{p81} and Lyons \cite{l86}. The
Rajchman property completely answers a question of Kahane and Salem
\cite{Kahane-Salem-64}, identifying $\mathscr N(\mathscr{B}, \mathscr{B}')$ as
a set of multiplicity (in the Fourier-analytic sense) if and only if
$(\mathscr{B}, \mathscr{B}')$ is compatible. The methodological contribution of
the article is the construction of a class of probability measures called
skewed measures. These measures depend on a number of parameters that can be
independently adjusted to ensure (subsets of) properties such as almost
everywhere normality, non-normality, ball conditions and Fourier decay.
