I am studying Moment generating functions from a Measure Theory perspective and I've found some difficulties with this exercise.
Let $X$ be a positive random variable on the probability space $(\Omega,\mathcal A,P)$. The function $\phi(t):=\int_{\Omega} e^{-tX} dP$ is the moment generating function. Now I am asked to show that the latter is $m$-times differentiable at $t=0+$ if the absolute moment $\int |X|^m dP<\infty$.
In my textbook the author simply states that :
$$|\frac {d^{m}}{dt^m} \phi(t)|=|X^m e^{-tX}|\leq X^m \color{red}*$$
And then, applying $m$-times Dominated Convergence he obtains that $$ \phi^{(m)}_X(0+)=(-1)^m\int X^mdP$$
$\color{red}*$ This would only hold true if $t\geq 0$ but, in general one assumes that $t\in\mathbb R$.
Is there some additional assumption I'm missing? Thanks in advance.