From PlanetMath and Wolfram, the Fourier-Stieltjes transform of a function $\alpha$ is defined as $\displaystyle \int_{\mathbb{R}} e^{itx} d(\alpha(t)).$ The kernel $\displaystyle e^{itx}$ is unlike the one $\displaystyle e^{-itx}$ used in Fourier transformation.
From Wikipedia, the Laplace-Stieltjes transform of a function $g$ is given as $ \displaystyle \int_{\mathbb{R}} e^{-sx}\,dg(x).$ The kernel ${e}^{-sx}$ is same as the one used in Laplace transformation.
I was wondering why the minus sign in the exponent of kernels for Fourier-Stieltjes transform and Fourier transform are not consistent, while consistent for Laplace-Stieltjes transform and Laplace transform?
Are there mistakes in the quoted sources, or other popular variant definitions?
I guess this is mostly a matter of convention. I think the minus convention in Fourier Series comes from the fact that
$\int_0^{2 \pi} f(t) \, e^{-int} \, dt$
Is the hermitian scalar product of $f$ and $\epsilon^n : x \mapsto e^{inx}$, which makes Fourier inversion formula more natural :
$f = \sum_{n \in \mathbb{Z}} \langle f \,|\, \epsilon^n \rangle \, \epsilon^n$
If you look at planetmath and wolfram, their point of view is to take a function $f$ and write it as an integral of $\epsilon$ with regards to a certain $d \alpha$. On the other hand, Wikipedia starts from $\alpha$ and then defines $f$. I guess the (admittedly subtle) difference in point of view might explain their convention.