I've been studying the Laplace-Stieltjes transform and I noticed that, according to the definition, the Laplace transform of any probability distribution at point $s = 0$ should be 1.
The transform for a random variable $X$ can be defined as:
$$X^*(s) = E[e^{-sX}]$$
Hence, at point $s = 0$, $X^*(0) = E[e^{-0X}] = E[1] = 1$.
This holds true for most distributions, except the uniform distribution. The Laplace-Stieltjes for the uniform distribution on the interval $(a,b)$ is given by:
$$U^*(s) = E[e^{-sU}] = \int_{a}^{b} e^{-su} \frac{1}{b-a}du = \frac{e^{-sa} - e^{-sb}}{s(b-a)}$$
Which at $s = 0$ is undefined (or so I think). My question is: why isn't it 1 (the definition leads me to believe it should be one)? Is there something incorrect with my train of thought?