Given a positive definite, measurable function $f(x)$ on the interval $(-\infty ,\infty)$ there exists a monotone increasing, real-valued bounded function $ \alpha (t)$ such that:

\begin{equation} f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t)), \end{equation} for all $x \in{\mathbb{R}}$ except a `small' set, that is a finite set which contains only a small number of values. When $f(x)$ is defined as above and if $\alpha(t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $\alpha(t)$ , and it is continuous in addition to being positive definite.