I'm working on the following exercise:
Let $X$ be a random variable on $(\Omega, \mathcal A, \mathbf P)$ and let $$\Lambda(t) := \log\left(\mathbf E\left[e^{tX}\right]\right) \quad \textrm{for all } t \in \mathbb R.$$ Show that $D := \{t \in \mathbb R : \Lambda(t) < \infty\}$ is a nonempty interval.
Clearly $\Lambda(0) = 0$ regardless of $X$, but I don't think it's generally true that $D$ is a nonempty interval. A counterexample could be $\Omega = \left[ -\frac 1 2, \frac 1 2 \right]$, $\mathbf P = \mathrm{Leb}$, and $X : \Omega \to \mathbb R$ is defined by $$ X(\omega) = \begin{cases} \frac 1 \omega & \textrm{if } \omega \neq 0, \\ 0 &\textrm{if } \omega = 0. \end{cases} $$ Then for $t > 0$, $$ \mathbf E\left[e^{tX}\right] \geq \int_0^{1/2} e^{t/\omega} \, d\omega = \int_{-1/2}^0 e^{-t/\omega} \, d\omega = \infty, $$ so $\Lambda(t) < \infty$ only for $t=0$. Unless we're defining $\{0\} = [0,0]$ as a nonempty interval with empty interior, this statement seems false in generally. Poking around online, I found lots of results with $\Lambda(t)$ being finite on $(-\delta, \delta)$ for some $\delta > 0$, but very little on when this hypothesis is satisfied.
What are the conditions on $X$ for there being an interval around the origin (or an interval containing the origin, such as $[0, \delta]$) such that $\Lambda(t) < \infty$?
See my comment to see that your example is not a counterexample. What is asserted is simply that $a<b<c$, $\Lambda (a) <\infty,\Lambda (b) <\infty $ imply $\Lambda (c) <\infty$ which follows from the fact that $e^{cX} <e^{aX}+e^{bX}$. No necessary and sufficient conditions for finiteness of $\Lambda$ in a neighbourhood of $0$ are available. If $X$ is a bounded random variable then $\Lambda (t)$ is finite in for all $t$. If $P\{X>s\} \leq a e^{-cs}$ for $s$ sufficently large, for some $a,c >0$ then $\Lambda (t)<\infty$ in a neighbothood of $0$.